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# Unknown Title ## Graphic lambda calculus Marius Buliga Institute of Mathematics, Romanian Academy P.O. BOX 1-764, RO 014700 Bucure¸ sti, Romania Marius.Buliga@imar.ro This version: 23.05.2013 ## Abstract We introduce and study graphic lambda calculus, a visual language which can be used for representing untyped lambda calculus, but it can also be used for computations in emergent algebras or for representing Reidemeister moves of locally planar tangle diagrams. ## 1 Introduction Graphic lambda calculus consists of a class of graphs endowed with moves between them. It might be considered a visual language in the sense of Erwig [9]. The name 'graphic lambda calculus' comes from the fact that it can be used for representing terms and reductions from untyped lambda calculus. It's main move is called 'graphic beta move' for it's relation to the beta reduction in lambda calculus. However, the graphic beta move can be applied outside the 'sector' of untyped lambda calculus, and the graphic lambda calculus can be used for other purposes than the one of visual representing lambda calculus. For other visual, diagrammatic representation of lambda calculus see the VEX language [8], or David Keenan's [15]. The motivation for introducing graphic lambda calculus comes from the study of emergent algebras. In fact, my goal is to build eventually a logic system which can be used for the formalization of certain 'computations' in emergent algebras, which can be applied then for a discrete differential calculus which exists for metric spaces with dilations, comprising riemannian manifolds and sub-riemannian spaces with very low regularity. Emergent algebras are a generalization of quandles, namely an emergent algebra is a family of idempotent right quasigroups indexed by the elements of an abelian group, while quandles are self-distributive idempotent right quasigroups. Tangle diagrams decorated by quandles or racks are a well known tool in knot theory [10] [13]. It is notable to mention the work of Kauffman [14], where the author uses knot diagrams for representing combinatory logic, thus untyped lambda calculus. Also Meredith and Snyder[17] associate to any knot diagram a process in pi-calculus, Is there any common ground between these three apparently separated field, namely differential calculus, logic and tangle diagrams? As a first attempt for understanding this, I proposed λ -Scale calculus [5], which is a formalism which contains both untyped lambda calculus and emergent algebras. Also, in the paper [6] I proposed a formalism of decorated tangle diagrams for emergent algebras and I called 'computing with space' the various manipulations of these diagrams with geometric content. Nevertheless, in that paper I was not able to give a precise sense of the use of the word 'computing'. I speculated, by using analogies from studies of the visual system, especially the 'Brain a geometry engine' paradigm of Koenderink [16], that, in order for the visual front end of the brain to reconstruct the visual space in the brain, there should be a kind of 'geometrical computation' in the neural network of the brain akin to the manipulation of decorated tangle diagrams described in our paper. I hope to convince the reader that graphic lambda calculus gives a rigorous answer to this question, being a formalism which contains, in a sense, lambda calculus, emergent algebras and tangle diagrams formalisms. Acknowledgement. This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, project number PN-II-ID-PCE-2011-30383. ## 2 Graphs and moves An oriented graph is a pair ( V, E ), with V the set of nodes and E ⊂ V × V the set of edges. Let us denote by α : V → 2 E the map which associates to any node N ∈ V the set of adjacent edges α ( N ). In this paper we work with locally planar graphs with decorated nodes, i.e. we shall attach to a graph ( V, E ) supplementary information: - -a function f : V → A which associates to any node N ∈ V an element of the 'graphical alphabet' A (see definition 2.1), - -a cyclic order of α ( N ) for any N ∈ V , which is equivalent to giving a local embedding of the node N and edges adjacent to it into the plane. We shall construct a set of locally planar graphs with decorated nodes, starting from a graphical alphabet of elementary graphs. On the set of graphs we shall define local transformations, or moves. Global moves or conditions will be then introduced. Definition 2.1 The graphical alphabet contains the elementary graphs, or gates, denoted by λ , Υ , , , and for any element ε of the commutative group Γ , a graph denoted by ¯ ε . Here are the elements of the graphical alphabet: <details> <summary>Image 1 Details</summary>  ### Visual Description \n ## Diagram: Graph Types ### Overview The image presents a collection of diagrams representing different graph types, each labeled with a Greek letter and the term "graph". The diagrams consist of nodes with arrows emanating from them, indicating directed edges. ### Components/Axes The image contains the following labeled graph types: * λ graph (appears twice) * Γ graph * Ξ graph * ε graph * Τ graph Each graph consists of a central node with three arrows pointing outwards. The Τ graph is an exception, consisting of a vertical line with an arrow pointing downwards. ### Detailed Analysis or Content Details 1. **λ graph (top-left):** A central node labeled "λ" with three arrows pointing outwards. 2. **Γ graph (top-right):** A central node labeled "Γ" with three arrows pointing outwards. 3. **λ graph (center-left):** A central node with a circle inside, and a line through it, labeled "λ" with three arrows pointing outwards. 4. **Ξ graph (center-right):** A central node labeled "Ξ" with three arrows pointing outwards. 5. **ε graph (bottom-right):** A central node labeled "ε" with three arrows pointing outwards. 6. **Τ graph (bottom-center):** A vertical line with an arrow pointing downwards, labeled "Τ". ### Key Observations All graphs except the Τ graph have a similar structure: a central node with three outgoing arrows. The Τ graph is a distinct, simpler structure. The λ graph appears twice, once with a simple label and once with a circle and line within the node. ### Interpretation The image likely illustrates different types of graphs used in a specific mathematical or computational context. The Greek letter labels suggest these graphs might represent specific functions, transformations, or relationships within a larger system. The variations in the central node (simple label vs. circle/line) could indicate different properties or constraints associated with each graph type. The Τ graph, being a simpler structure, might represent a base case or a fundamental operation. Without further context, it's difficult to determine the precise meaning of each graph type, but the image clearly aims to categorize and visually distinguish them. The image does not provide any quantitative data, but rather a qualitative representation of different graph structures. </details> With the exception of the , all other elementary graphs have three edges. The graph has only one edge. There are two types of 'fork' graphs, the λ graph and the Υ graph, and two types of 'join' graphs, the graph and the ¯ ε graph. Further I briefly explain what are they supposed to represent and why they are needed in this graphic formalism. The λ gate corresponds to the lambda abstraction operation from untyped lambda calculus. This gate has one input (the entry arrow) and two outputs (the exit arrows), therefore, at first view, it cannot be a graphical representation of an operation. In untyped lambda calculus the λ abstraction operation has two inputs, namely a variable name x and a term A , and one output, the term λx.A . There is an algorithm, presented in section 3, which transforms a lambda calculus term into a graph made by elementary gates, such that to any lambda abstraction which appears in the term corresponds a λ gate. The Υ gate corresponds to a FAN-OUT gate. It is needed because the graphic lambda calculus described in this article does not have variable names. Υgates appear in the process of elimination of variable names from lambda terms, in the algorithm previously mentioned. Another justification for the existence of two fork graphs is that they are subjected to different moves: the λ gate appears in the graphic beta move, together with the gate, while the Υ gate appears in the FAN-OUT moves. Thus, the λ and Υ gates, even if they have the same topology, they are subjected to different moves, which in fact characterize their 'lambda abstraction'-ness and the 'fan-out'-ness of the respective gates. The alternative, which consists into using only one, generic, fork gate, leads to the identification, in a sense, of lambda abstraction with fan-out, which would be confusing. The gate corresponds to the application operation from lambda calculus. The algorithm from section 3 associates a gate to any application operation used in a lambda calculus term. The ¯ ε gate corresponds to an idempotent right quasigroup operation, which appears in emergent algebras, as an abstractization of the geometrical operation of taking a dilation (of coefficient ε ), based at a point and applied to another point. As previously, the existence of two join gates, with the same topology, is justified by the fact that they appear in different moves. 1. The set GRAPH. We construct the set of graphs GRAPH over the graphical alphabet by grafting edges of a finite number of copies of the elements of the graphical alphabet. Definition 2.2 GRAPH is the set of graphs obtained by grafting edges of a finite number of copies of the elements of the graphical alphabet. During the grafting procedure, we start from a set of gates and we add, one by one, a finite number of gates, such that, at any step, any edge of any elementary graph is grafted on any other free edge (i.e. not already grafted to other edge) of the graph, with the condition that they have the same orientation. For any node of the graph, the local embedding into the plane is given by the element of the graphical alphabet which decorates it. The set of free edges of a graph G ∈ GRAPH is named the set of leaves L ( G ) . Technically, one may imagine that we complete the graph G ∈ GRAPH by adding to the free extremity of any free edge a decorated node, called 'leaf', with decoration 'IN' or 'OUT', depending on the orientation of the respective free edge. The set of leaves L ( G ) thus decomposes into a disjoint union L ( G ) = IN ( G ) ∪ OUT ( G ) of in or out leaves. Moreover, we admit into GRAPH arrows without nodes, , called wires or lines, and loops (without nodes from the elementary graphs, nor leaves) <details> <summary>Image 2 Details</summary>  ### Visual Description \n ## Diagram: Circular Flow with Arrow ### Overview The image depicts a simple diagram consisting of a circle with an arrow indicating a clockwise flow. There are no labels, axes, or legends present. The diagram appears to represent a cyclical process or system. ### Components/Axes There are no explicit components, axes, or legends. The diagram consists solely of: * A circle, defined by a continuous black line. * An arrow, also in black, positioned along the circumference of the circle, pointing in a clockwise direction. ### Detailed Analysis or Content Details The circle is approximately elliptical, slightly wider than it is tall. The arrow originates from the upper-right quadrant of the circle and curves clockwise along the circumference. The arrow's head is clearly defined, indicating the direction of flow. There are no numerical values or specific data points within the diagram. ### Key Observations The diagram's simplicity suggests a fundamental or abstract concept. The clockwise arrow implies a continuous, repeating process. The lack of labels or context makes it difficult to determine the specific process being represented. ### Interpretation The diagram likely represents a cyclical process, feedback loop, or system where the output returns to the input. Without further context, it's impossible to determine the nature of this cycle. It could represent anything from a simple iterative process to a complex biological or economic system. The diagram's abstract nature suggests it's intended to convey a general principle rather than a specific instance. The arrow's direction indicates a positive feedback loop, where the process reinforces itself. The absence of any other elements suggests that the cycle is self-contained and doesn't have external inputs or outputs. </details> Graphs in GRAPH can be disconnected. Any graph which is a finite reunion of lines, loops and assemblies of the elementary graphs is in GRAPH . 2. Local moves. These are transformations of graphs in GRAPH which are local, in the sense that any of the moves apply to a limited part of a graph, keeping the rest of the graph unchanged. We may define a local move as a rule of transformation of a graph into another of the following form. First, a subgraph of a graph G in GRAPH is any collection of nodes and/or edges of G . It is not supposed that the mentioned subgraph must be in GRAPH . Also, a collection of some edges of G , without any node, count as a subgraph of G . Thus, a subgraph of G might be imagined as a subset of the reunion of nodes and edges of G . For any natural number N and any graph G in GRAPH , let P ( G,N ) be the collection of subgraphs P of the graph G which have the sum of the number of edges and nodes less than or equal to N . Definition 2.3 A local move has the following form: there is a number N and a condition C which is formulated in terms of graphs which have the sum of the number of edges and nodes less than or equal to N , such that for any graph G in GRAPH and for any P ∈ P ( G,N ) , if C is true for P then transform P into P ′ , where P ′ is also a graph which have the sum of the number of edges and nodes less than or equal to N . Graphically we may group the elements of the subgraph, subjected to the application of the local rule, into a region encircled with a dashed closed, simple curve. The edges which cross the curve (thus connecting the subgraph P with the rest of the graph) will be numbered clockwise. The transformation will affect only the part of the graph which is inside the dashed curve (inside meaning the bounded connected part of the plane which is bounded by the dashed curve) and, after the transformation is performed, the edges of the transformed graph will connect to the graph outside the dashed curve by respecting the numbering of the edges which cross the dashed line. However, the grouping of the elements of the subgraph has no intrinsic meaning in graphic lambda calculus. It is just a visual help and it is not a part of the formalism. As a visual help, I shall use sometimes colors in the figures. The colors, as well, don't have any intrinsic meaning in the graphic lambda calculus. 2.1. Graphic β move. This is the most important move, inspired by the β -reduction from lambda calculus, see theorem 3.1, part (d). <details> <summary>Image 3 Details</summary>  ### Visual Description \n ## Diagram: Causal Network Representation ### Overview The image depicts two causal network diagrams. The diagram on the left shows a more complex network with two decision nodes, while the diagram on the right shows a simpler network with direct causal links. Both diagrams use arrows to indicate the direction of causal influence. ### Components/Axes The diagrams consist of nodes (circles or points) representing variables and arrows representing causal relationships. The left diagram includes labels "λ" and "β" associated with the decision nodes. The nodes are numbered 1 through 4. ### Detailed Analysis or Content Details **Left Diagram:** * **Node 1:** Has a directed arrow pointing *into* the top decision node (labeled "λ"). * **Node 2:** Has a directed arrow pointing *out* of the top decision node (labeled "λ"). * **Node 3:** Has a directed arrow pointing *out* of the bottom decision node. * **Node 4:** Has a directed arrow pointing *into* the bottom decision node. * **Decision Node (Top):** Labeled "λ". Receives input from Node 1 and has an output to Node 2. * **Decision Node (Bottom):** Receives input from Node 4 and has an output to Node 3. * The two decision nodes are connected by a downward arrow, indicating a causal relationship. **Right Diagram:** * **Node 1:** Has a directed arrow pointing to Node 3. * **Node 4:** Has a directed arrow pointing to Node 2. * **Bidirectional Arrow:** A blue, double-headed arrow labeled "β" connects Node 1 and Node 4. This indicates a reciprocal causal relationship between these two nodes. ### Key Observations The left diagram represents a more complex causal structure with two decision points and a connection between them. The right diagram shows a simpler structure with direct causal links and a bidirectional relationship between nodes 1 and 4. The use of "λ" and "β" suggests these are parameters or variables within the causal models. ### Interpretation The diagrams illustrate causal relationships between variables. The left diagram suggests that Node 1 and Node 4 influence Nodes 2 and 3 respectively, through decision nodes labeled "λ" and an unspecified node. The connection between the decision nodes implies a dependency or interaction between the two processes. The right diagram shows a simpler system where Node 1 influences Node 3 and Node 4 influences Node 2, with a reciprocal influence between Node 1 and Node 4 represented by "β". These diagrams are likely used to model a system where variables influence each other, and the arrows represent the direction of causality. The labels "λ" and "β" could represent parameters or variables that quantify the strength or nature of these causal relationships. The diagrams could be part of a larger causal model used for prediction, intervention, or understanding the underlying mechanisms of a system. The diagrams do not provide any numerical data, but rather a qualitative representation of causal relationships. </details> The labels '1, 2, 3, 4' are used only as guides for gluing correctly the new pattern, after removing the old one. As with the encircling dashed curve, they have no intrinsic meaning in graphic lambda calculus. This 'sewing braids' move will be used also in contexts outside of lambda calculus! It is the most powerful move in this graphic calculus. A primitive form of this move appears as the re-wiring move (W1) (section 3.3, p. 20 and the last paragraph and figure from section 3.4, p. 21 in [6]). An alternative notation for this move is the following: <details> <summary>Image 4 Details</summary>  ### Visual Description \n ## Diagram: Bifurcation Diagram ### Overview The image presents two diagrams illustrating a bifurcation process. The left diagram depicts a branching structure with labeled nodes and arrows, while the right diagram shows a simplified crossing of lines representing the same process. A bidirectional arrow labeled "β" connects the two diagrams. ### Components/Axes The diagrams utilize the following components: * **Nodes:** Circular nodes with internal patterns (circles within circles). * **Arrows:** Arrows indicating flow or connection between nodes/lines. * **Labels:** Numbers 1 through 4 labeling the input/output points. * **Label:** "λ" inside the top circular node. * **Label:** "β" labeling the bidirectional arrow. ### Detailed Analysis or Content Details **Left Diagram:** * Input 1 splits into two outputs: 2 and a connection to the lower node. * Input 4 splits into two outputs: 3 and a connection to the lower node. * The lower node has a pattern of nested circles. * The label "λ" is positioned inside the top circular node. **Right Diagram:** * Line originating from point 1 crosses with a line originating from point 4. * The crossing creates two output lines, one leading to point 2 and the other to point 3. * Arrows indicate the direction of flow along each line. **Bidirectional Arrow:** * The arrow labeled "β" connects the two diagrams, indicating a relationship or transformation between them. The arrow has two heads, indicating the relationship is bidirectional. ### Key Observations * The left diagram is a more detailed representation of the branching process, while the right diagram is a simplified schematic. * The label "λ" might represent a parameter or condition influencing the branching. * The label "β" likely represents a transformation or relationship between the detailed and simplified representations of the bifurcation. * The diagrams do not contain numerical data or scales. ### Interpretation The diagrams illustrate a bifurcation, a point where a system's behavior splits into two or more possibilities. The left diagram shows a more complex branching structure, potentially representing a detailed model of the bifurcation process. The right diagram provides a simplified, schematic view of the same process, focusing on the crossing of paths. The "λ" label likely represents a control parameter that influences the branching, while "β" represents the transformation or mapping between the detailed and simplified representations. The bidirectional arrow suggests that the simplified model can be used to understand the detailed model, and vice versa. This type of diagram is common in dynamical systems theory, particularly when analyzing how systems change their behavior as parameters are varied. The absence of numerical data suggests the diagrams are conceptual rather than quantitative. </details> A move which looks very much alike the graphic beta move is the UNZIP operation from the formalism of knotted trivalent graphs, see for example the paper [21] section 3. In order to see this, let's draw again the graphic beta move, this time without labeling the arrows: <details> <summary>Image 5 Details</summary>  ### Visual Description \n ## Diagram: Feynman Diagram Representation ### Overview The image depicts a Feynman diagram, a pictorial representation of particle interactions in quantum field theory. It shows a process involving a particle emitting and then absorbing a photon, represented by the wavy line. A second diagram shows two particles crossing. The diagrams are separated by a symbol representing beta. ### Components/Axes The diagram consists of the following components: * **Lines:** Straight lines represent fermions (matter particles). Arrows on the lines indicate the direction of particle propagation (time flow). * **Wavy Line:** A wavy line represents a photon (force carrier). * **Vertices:** Circles represent interaction vertices where particles interact. * **β:** A symbol, likely representing a parameter or angle, with a blue, curved, double-headed arrow. ### Detailed Analysis or Content Details The first diagram shows: * A single fermion line entering a vertex. * From this vertex, a wavy line (photon) emerges. * The photon then enters another vertex. * From the second vertex, multiple fermion lines exit. * The first vertex has a symbol resembling a sine wave (λ) inside the circle. The second diagram shows: * Two straight lines crossing each other. * Each line has an arrow indicating the direction of particle flow. * The lines intersect at an angle. The symbol β is positioned between the two diagrams. It consists of a curved, double-headed arrow in blue. ### Key Observations * The first diagram represents a process where a particle emits a photon and then interacts with other particles. * The second diagram represents a scattering process where two particles exchange momentum. * The symbol β likely represents a physical quantity related to the interaction, such as a scattering angle or a coupling constant. * The diagrams are simplified representations of complex quantum processes. ### Interpretation The image illustrates fundamental concepts in quantum field theory. The Feynman diagrams provide a visual way to understand particle interactions. The first diagram shows a basic electromagnetic interaction, where a charged particle emits and absorbs a photon. The second diagram shows a scattering event, where two particles interact and change their trajectories. The symbol β suggests that the diagrams are related to a specific physical process or calculation. The diagrams are not providing numerical data, but rather a qualitative representation of particle behavior. The diagrams are a visual language for physicists to describe and calculate the probabilities of different particle interactions. The use of arrows indicates the direction of time and particle flow, which is crucial for understanding the dynamics of these interactions. </details> The unzip operation acts only from left to right in the following figure. Remarkably, it acts on trivalent graphs (but not oriented). <details> <summary>Image 6 Details</summary>  ### Visual Description \n ## Diagram: Molecular Unzipping ### Overview The image depicts a simplified diagram illustrating a molecular "unzipping" process. It shows a molecule with a central connection point transforming into a linear structure. The diagram is schematic and does not contain numerical data. ### Components/Axes The diagram consists of the following elements: * **Initial Molecule:** A structure resembling a central connection point with four lines extending outwards at approximately 45-degree angles. The connection points are represented by filled black circles. * **"UNZIP" Label:** The word "UNZIP" is written in blue, positioned between the initial molecule and the resulting linear structure. * **Arrow:** A curved blue arrow indicates the direction of the transformation, pointing from the initial molecule towards the linear structure. * **Final Structure:** Two parallel black lines representing the "unzipped" molecule. ### Detailed Analysis or Content Details The initial molecule appears to be a simplified representation of a molecular structure with a central bond. The "UNZIP" label and the arrow suggest a process where this bond is broken, resulting in a linear arrangement of the molecular components. The final structure consists of two parallel lines, indicating a separation or unfolding of the initial molecule. There are no quantitative values or scales present in the diagram. ### Key Observations The diagram visually represents a process of molecular separation or unfolding. The use of the term "UNZIP" suggests a breaking of a central connection, leading to a linear arrangement. The simplicity of the diagram implies a conceptual illustration rather than a precise depiction of a specific molecular structure. ### Interpretation The diagram illustrates a conceptual process of molecular unzipping. This could represent the breaking of a chemical bond, the separation of DNA strands, or a similar process where a connected structure is transformed into a linear one. The diagram is not specific to any particular molecule or reaction, but rather serves as a general illustration of a separation process. The use of the term "UNZIP" evokes the imagery of a zipper being opened, suggesting a relatively straightforward and reversible process. The lack of detail suggests that the diagram is intended to convey a general principle rather than a specific scientific observation. </details> Let us go back to the graphic beta move and remark that it does not depend on the particular embedding in the plane. For example, the intersection of the '1,3' arrow with the '4,2' arrow is an artifact of the embedding, there is no node there. Intersections of arrows have no meaning, remember that we work with graphs which are locally planar, not globally planar. The graphic beta move goes into both directions. In order to apply the move, we may pick a pair of arrows and label them with '1,2,3,4', such that, according to the orientation of the arrows, '1' points to '3' and '4' points to '2', without any node or label between '1' and '3' and between '4' and '2' respectively. Then, by a graphic beta move, we may replace the portions of the two arrows which are between '1' and '3', respectively between '4' and '2', by the pattern from the LHS of the figure. The graphic beta move may be applied even to a single arrow, or to a loop. In the next figure we see three applications of the graphic beta move. They illustrate the need for considering loops and wires as members of GRAPH . <details> <summary>Image 7 Details</summary>  ### Visual Description \n ## Diagram: Permutation Group Transformations ### Overview The image presents a series of diagrams illustrating transformations between linear permutations and cycle decompositions of permutations. Each row depicts a permutation in both linear notation and cycle notation, connected by a blue, curved arrow labeled "β". The diagrams show how a permutation can be represented as a sequence of elements and as a set of cycles. ### Components/Axes The image consists of three rows, each containing a linear permutation, a transformation arrow labeled "β", and a corresponding cycle decomposition diagram. - **Linear Permutations:** Represented as horizontal lines with numbers 1 through 4 arranged in a specific order. - **Transformation Arrow (β):** A curved blue arrow indicating the transformation from linear notation to cycle notation. - **Cycle Decomposition Diagrams:** Represented as nodes and directed edges, illustrating the cycles within the permutation. Nodes are labeled with numbers 1 through 4. The symbol "λ" and a "Y" shape are used to represent the branching of cycles. ### Detailed Analysis or Content Details **Row 1:** - **Linear Permutation:** 1 3 4 2. This means 1 maps to 3, 3 maps to 4, 4 maps to 2, and 2 maps to 1. - **Cycle Decomposition:** The diagram shows a cycle (1 3 4 2). This is represented by a circular arrangement of nodes 1, 3, 4, and 2, with directed edges connecting them in that order, and an edge returning from 2 to 1. There is a node labeled "λ" with an arrow pointing to node 1, and a "Y" shape connecting nodes 1, 3, 4, and 2. **Row 2:** - **Linear Permutation:** 4 2 1 3. This means 4 maps to 2, 2 maps to 1, 1 maps to 3, and 3 maps to 4. - **Cycle Decomposition:** The diagram shows a cycle (4 2 1 3). This is represented by a circular arrangement of nodes 4, 2, 1, and 3, with directed edges connecting them in that order, and an edge returning from 3 to 4. There is a node labeled "λ" with an arrow pointing to node 4, and a "Y" shape connecting nodes 4, 2, 1, and 3. **Row 3:** - **Linear Permutation:** 1 2 3 4. This represents the identity permutation, where each element maps to itself. - **Cycle Decomposition:** The diagram shows a single cycle (1 2 3 4). This is represented by a circular arrangement of nodes 1, 2, 3, and 4, with directed edges connecting them in that order, and an edge returning from 4 to 1. There is a node labeled "λ" with an arrow pointing to node 1, and a "Y" shape connecting nodes 1, 2, 3, and 4. ### Key Observations - The transformation "β" consistently maps a linear permutation to its equivalent cycle decomposition. - The cycle decomposition diagrams visually represent the cycles within each permutation. - The number of cycles in the decomposition corresponds to the number of disjoint cycles in the permutation. - The identity permutation (1 2 3 4) is represented by a single cycle containing all elements. ### Interpretation The diagrams illustrate the fundamental relationship between linear permutations and cycle decompositions in permutation group theory. The transformation "β" represents the process of expressing a permutation as a product of disjoint cycles. This representation is often more concise and insightful for understanding the permutation's behavior. The "λ" and "Y" shapes in the cycle decomposition diagrams likely represent a branching point or a way to visually emphasize the cyclic nature of the permutation. The diagrams demonstrate how permutations can be broken down into simpler, cyclic components, which is crucial for analyzing their properties and applying them in various mathematical and computational contexts. The diagrams are a visual aid for understanding the structure of permutations and how they operate on sets of elements. </details> Also, we can apply in different ways a graphic beta move, to the same graph and in the same place, simply by using different labels '1', ... '4' (here A , B , C , D are graphs in GRAPH ): A particular case of the previous figure is yet another justification for having loops as elements in GRAPH . <details> <summary>Image 8 Details</summary>  ### Visual Description \n ## Diagram: State Transition/Flow Diagram ### Overview The image presents two pairs of diagrams illustrating state transitions or flow between four entities labeled A, B, C, and D. Each pair shows an initial state on the left and a transformed state on the right, connected by a bidirectional arrow labeled "β". The transitions are numerically labeled from 1 to 4. The diagrams utilize a node-and-arrow structure to represent the flow. ### Components/Axes The diagrams consist of: * **Nodes:** Represented by pentagonal shapes labeled A, B, C, and D. * **Arrows:** Indicate the direction of flow or transition. * **Numerical Labels:** Numbers 1 through 4 along the arrows, denoting the order or identifier of the transition. * **Bidirectional Arrow:** Labeled "β", indicating a transformation or relationship between the two diagrams in each pair. * **Circular Node:** A circle with a branching arrow, representing a decision or split point in the flow. ### Detailed Analysis or Content Details **Diagram Pair 1 (Top)** * **Left Diagram:** * A connects to a circular node via arrow labeled '1'. * D connects to the same circular node via arrow labeled '4'. * The circular node splits into two arrows: one to C labeled '2', and one to B labeled '3'. * **Right Diagram:** * A connects to B via arrow labeled '3'. * D connects to C via arrow labeled '2'. * A connects to B via arrow labeled '3'. * D connects to C via arrow labeled '4'. **Diagram Pair 2 (Bottom)** * **Left Diagram:** * D connects to a circular node via arrow labeled '1'. * A connects to the same circular node via arrow labeled '4'. * The circular node splits into two arrows: one to B labeled '2', and one to C labeled '3'. * **Right Diagram:** * A connects to B via arrow labeled '2'. * D connects to C via arrow labeled '3'. * A connects to B via arrow labeled '4'. * D connects to C via arrow labeled '1'. ### Key Observations * The diagrams demonstrate a rearrangement of connections between the entities A, B, C, and D. * The "β" arrow suggests a transformation or equivalence between the initial and transformed states. * The numerical labels on the arrows appear to be consistently reassigned between the left and right diagrams within each pair. * The circular node acts as a branching point, distributing flow to multiple entities. ### Interpretation The diagrams likely represent a reversible transformation or a duality in the relationships between the entities A, B, C, and D. The "β" arrow indicates that the two diagrams are equivalent under some operation. The numerical labels on the arrows suggest a mapping or permutation of connections. The circular node could represent a decision point or a common intermediate state. The transformation appears to involve swapping the connections between A/D and B/C. In the first pair, A and D initially connect to a branching point that leads to C and B. After the transformation (β), A connects directly to B, and D connects directly to C. The second pair shows a similar transformation, but with the initial connections reversed. This suggests a symmetry or duality in the system. The diagrams could be illustrating a concept from graph theory, network analysis, or a similar field where relationships between entities are important. Without further context, it's difficult to determine the specific meaning of the transformation "β" or the significance of the numerical labels. </details> These two applications of the graphic beta move may be represented alternatively like this: <details> <summary>Image 9 Details</summary>  ### Visual Description \n ## Diagram: Transformation of Graph Structures ### Overview The image depicts two sets of graph transformations. Each set shows an initial complex graph structure being transformed into a simpler, linear structure via a process labeled "β". The graphs consist of nodes represented by shapes (circles and pentagons) and directed edges labeled with numerical values. ### Components/Axes The diagram consists of the following components: * **Nodes:** Represented by circles and pentagons. The pentagons are labeled "C" and "D". The circles are unlabeled, but appear to represent branching points. * **Edges:** Directed lines connecting the nodes, labeled with the numbers 1, 2, 3, and 4. * **Transformation Arrows:** Blue, curved arrows labeled "β" indicating the transformation process. * **Enclosing Curves:** Red and blue curves that appear to delineate the initial and transformed graph structures. ### Detailed Analysis or Content Details **First Transformation (Top):** * **Initial Graph:** A central node (circle) has incoming edges labeled '1' from a pentagon 'D' and '2' from a pentagon 'C'. It has outgoing edges labeled '3' to a pentagon 'C' and '4' to a pentagon 'D'. * **Transformed Graphs (Right):** * A linear sequence with edges labeled '1' and '3'. * A linear sequence with edges labeled '4' and '2'. * A linear sequence with edges labeled '1' and '3', with 'D' and 'C' pentagons. **Second Transformation (Bottom):** * **Initial Graph:** A central node (circle) has incoming edges labeled '1' from a pentagon 'D' and '2' from a pentagon 'C'. It has outgoing edges labeled '3' to a pentagon 'C' and '4' to a pentagon 'D'. This is identical to the first initial graph. * **Transformed Graphs (Right):** * A linear sequence with edges labeled '4' and '2'. * A linear sequence with edges labeled '1' and '3', with 'D' and 'C' pentagons. ### Key Observations * The initial graph structure in both transformations is identical. * The transformation "β" consistently breaks down the complex graph into simpler linear sequences. * The edges are re-arranged, but the labels (1, 2, 3, 4) remain consistent throughout the transformation. * The pentagons 'C' and 'D' are consistently present in the initial and transformed graphs. ### Interpretation The diagram likely represents a decomposition or simplification process applied to a graph structure. The transformation "β" could represent a rule or algorithm that breaks down a complex network into simpler paths. The consistent labeling of the edges suggests that the transformation preserves the relationships between the nodes, even as the overall structure changes. The presence of 'C' and 'D' could indicate specific types of nodes or entities within the network. The diagram demonstrates a potential method for reducing complexity in a graph by isolating specific pathways. The repetition of the initial graph and transformation suggests a general principle or rule being illustrated, rather than a specific instance. The transformation appears to separate the incoming and outgoing connections into distinct linear sequences. This could be a step in a larger process, such as pathfinding or network analysis. The enclosing curves may represent boundaries or scopes of the transformation. </details> <details> <summary>Image 10 Details</summary>  ### Visual Description \n ## Diagram: Reaction Pathway Illustration ### Overview The image presents a series of diagrams illustrating a chemical reaction pathway. The diagrams depict molecules (labeled D and C) and a transition state (labeled λ). Arrows indicate the flow of the reaction, and numbers are used to label specific steps or bonds. A double-headed arrow labeled "β" connects pairs of diagrams, suggesting an equilibrium or reversible process. ### Components/Axes The diagrams consist of the following components: * **Molecules:** Represented by stylized shapes, labeled "D" and "C". * **Transition State:** Represented by a double-X shape, labeled "λ". * **Arrows:** Indicate the direction of the reaction. * **Numbers:** Used to label bonds or steps in the reaction (1, 2, 3, 4). * **Double-Headed Arrow:** Labeled "β", indicating a reversible process or equilibrium. ### Detailed Analysis or Content Details The image contains four distinct diagrams, arranged in two pairs connected by the "β" arrow. **Top Pair:** * **Left Diagram:** Molecule D (labeled 4) reacts with the transition state λ (labeled 1 and 3), which then leads to molecule C (labeled 2). * **Right Diagram:** A loop is formed from molecule D (labeled 1 and 3). Molecule D (labeled 4) reacts to form molecule C (labeled 2). **Bottom Pair:** * **Left Diagram:** Molecule D (labeled 1) reacts with the transition state λ (labeled 2 and 4), which then leads to molecule C (labeled 3). * **Right Diagram:** A loop is formed from molecule C (labeled 2 and 4). Molecule D (labeled 1) reacts to form molecule C (labeled 3). ### Key Observations The diagrams illustrate two different reaction pathways, each with a reversible step represented by the "β" arrow. The numbers associated with the arrows and molecules suggest changes in bond formation or breaking during the reaction. The transition state (λ) appears to be a key intermediate in both pathways. The diagrams show a possible interconversion between the two pathways. ### Interpretation The diagrams likely represent a simplified model of a chemical reaction involving molecules D and C, with a transition state λ. The "β" arrow suggests that the reaction is reversible or that there is an equilibrium between the two pathways. The numbers likely represent specific bonds or interactions that are formed or broken during the reaction. The diagrams could be used to illustrate the mechanism of a chemical reaction, or to compare the relative rates of different pathways. The loops in the right diagrams suggest self-catalysis or internal rearrangement within the molecules. The diagrams are schematic and do not provide quantitative data, but they offer a qualitative understanding of the reaction process. The image does not provide any facts or data, but rather a visual representation of a theoretical process. </details> - 2.2. (CO-ASSOC) move. This is the 'co-associativity' move involving the Υ graphs. We think about the Υ graph as corresponding to a FAN-OUT gate. <details> <summary>Image 11 Details</summary>  ### Visual Description \n ## Diagram: Co-Association Schematic ### Overview The image presents a schematic diagram illustrating a co-association relationship between two identical branching structures. Each structure consists of two branching points, with input and output lines labeled 1 through 4. A blue, bidirectional arrow labeled "CO-ASSOC" connects the two structures, indicating a co-association link. ### Components/Axes The diagram consists of: * Two identical branching structures. * Four labeled input/output lines per structure (1, 2, 3, 4). * Branching points represented by circular nodes with three lines converging/diverging. * A bidirectional arrow labeled "CO-ASSOC" connecting the two structures. ### Detailed Analysis or Content Details The diagram depicts two identical structures. Each structure has a single input line labeled "1" at the bottom, which splits into two output lines. These output lines then converge at a second branching point, resulting in two further output lines labeled "3" and "4". The input line "2" enters the first branching point directly. The "CO-ASSOC" arrow is positioned horizontally between the two structures, approximately in the center. The arrow is blue and has a double-headed design, indicating a bidirectional relationship. The lines are all black and of similar thickness. The labels (1, 2, 3, 4, and CO-ASSOC) are also black. ### Key Observations * The diagram emphasizes symmetry between the two structures. * The "CO-ASSOC" label suggests a correlation or association between the two branching processes. * The diagram does not provide any quantitative data or specific values; it is a purely schematic representation. ### Interpretation The diagram likely represents a simplified model of a biological or computational process where two similar pathways or systems are interconnected and influence each other. The "CO-ASSOC" label suggests that the activity or state of one system is correlated with the activity or state of the other. The branching points could represent decision points or regulatory elements within the systems. The lack of quantitative data suggests that the diagram is intended to convey a conceptual relationship rather than precise measurements. The symmetry of the structures implies that the relationship is reciprocal or balanced. The diagram could be used to illustrate concepts in systems biology, network theory, or signal processing. It is a visual representation of a relationship, not a data-driven chart. </details> By using CO-ASSOC moves, we can move between any two binary trees formed only with Υ gates, with the same number of output leaves. - 2.3. (CO-COMM) move. This is the 'co-commutativity' move involving the Υ gate. It will be not used until the section 6 concerning knot diagrams. - 2.3.a (R1a) move. This move is imported from emergent algebras. Explanations are given in section 5. It involves an Υ graph and a ¯ ε graph, with ε ∈ Γ. <details> <summary>Image 12 Details</summary>  ### Visual Description \n ## Diagram: Communication Models ### Overview The image presents two diagrams illustrating different communication models. The diagram on the left depicts a circular, feedback-oriented model, while the diagram on the right shows a linear, one-way communication flow. A bidirectional arrow labeled "CO-COMM" connects the two diagrams, suggesting a relationship or transition between the two models. ### Components/Axes The diagrams utilize numbered lines (1, 2, and 3) to represent communication elements. Each diagram features a central node represented by a circle with a 'Y' shape inside, likely symbolizing a communicator or source. Arrows indicate the direction of communication flow. The label "CO-COMM" is positioned centrally between the two diagrams. ### Detailed Analysis or Content Details **Left Diagram (Circular Model):** * **Line 1:** An upward-pointing arrow originates from below the central node. * **Line 2:** A curved line originates from the top-left of the central node, looping around and intersecting with Line 3. * **Line 3:** A curved line originates from the top-right of the central node, looping around and intersecting with Line 2. * The intersection of Lines 2 and 3 forms a closed loop, indicating feedback. Arrows within the loop suggest a continuous flow of information. **Right Diagram (Linear Model):** * **Line 1:** An upward-pointing arrow originates from below the central node. * **Line 2:** A straight line extends upwards and to the left from the central node. * **Line 3:** A straight line extends upwards and to the right from the central node. * The lines 2 and 3 diverge from the central node, indicating a one-way flow of information. **Central Label:** * "CO-COMM" – This label is written in all caps and is positioned horizontally between the two diagrams. ### Key Observations The left diagram emphasizes a cyclical communication process with feedback, while the right diagram illustrates a linear, one-way communication process. The "CO-COMM" label suggests a connection or potential transition between these two models. The diagrams are simple and schematic, focusing on the flow of communication rather than specific content. ### Interpretation The diagrams likely represent contrasting models of communication. The left diagram, with its circular flow and feedback loop, could represent a more interactive or collaborative communication model. The right diagram, with its linear flow, could represent a more traditional, one-way communication model (e.g., a lecture or broadcast). The "CO-COMM" label might indicate "co-communication" or a combined communication approach, suggesting that communication can move between these two models depending on the context. The diagrams are abstract and do not provide specific data points, but they offer a visual representation of different communication dynamics. The 'Y' shape within the central node could represent the branching of communication channels or the multiple roles of a communicator. The lack of further context makes a definitive interpretation challenging, but the diagrams clearly highlight the difference between feedback-driven and linear communication processes. </details> <details> <summary>Image 13 Details</summary>  ### Visual Description \n ## Diagram: Flow Representation with Symbols ### Overview The image depicts a diagram illustrating a flow or transformation between two states. The left side shows a configuration with two circular nodes connected by arrows, while the right side shows a single, straight line with arrows. A curved blue arrow with the label "Rla" connects the two configurations, indicating a transformation. ### Components/Axes The diagram consists of the following components: * **Circular Nodes:** Two circular nodes, labeled "ε" (epsilon) and "Y" (gamma). * **Arrows:** Arrows indicating flow or direction. These are present within the circular node configuration and as a curved arrow connecting the two configurations. * **Label:** "Rla" – a label associated with the curved arrow. * **Straight Line:** A straight line with arrows indicating flow. ### Detailed Analysis or Content Details The left side of the diagram shows two circular nodes. The top node is labeled "ε" and the bottom node is labeled "Y". Arrows flow into and out of both nodes, creating a loop-like structure. Specifically: * Arrows enter the "ε" node from the left. * Arrows exit the "ε" node to the right and downwards towards the "Y" node. * Arrows enter the "Y" node from above (from the "ε" node). * Arrows exit the "Y" node to the left. The right side of the diagram shows a single straight line with arrows indicating flow from bottom-left to top-right. The curved blue arrow labeled "Rla" originates approximately from the center of the "ε" node and points towards the straight line, indicating a transformation or mapping from the circular node configuration to the straight line configuration. ### Key Observations The diagram visually represents a transformation from a complex, looped system (represented by the circular nodes) to a simpler, linear system (represented by the straight line). The label "Rla" likely denotes the specific operation or rule governing this transformation. The diagram does not contain any numerical data or scales. ### Interpretation The diagram likely represents a simplification or reduction process. The "Rla" transformation could be a rule or operation that removes the complexity represented by the looped system of "ε" and "Y", resulting in a more straightforward flow. The symbols "ε" and "Y" could represent variables, states, or components within a larger system. Without further context, the precise meaning of the diagram remains speculative, but it clearly illustrates a change in system structure or state. The diagram suggests a process of abstraction or simplification, where a complex interaction is reduced to a single flow. The use of Greek letters suggests a mathematical or scientific context. </details> 2.3.b (R1b) move. The move R1b (also related to emergent algebras) is this: <details> <summary>Image 14 Details</summary>  ### Visual Description \n ## Diagram: System State Transition ### Overview The image depicts a diagram illustrating a system state transition. It shows a complex state on the left, a transition arrow in the center, and a simpler state on the right. The diagram appears to represent a simplification or reduction of complexity within a system. ### Components/Axes The diagram consists of the following components: * **Left State:** A complex state represented by a series of interconnected lines and two circular nodes. The top node contains the symbol "Y", and the bottom node contains the symbol "ε" (epsilon). * **Transition Arrow:** A curved, blue arrow pointing from the left state to the right state. The arrow is labeled "R1b". * **Right State:** A simple state represented by a straight line with arrowheads at both ends. ### Detailed Analysis or Content Details The left state is characterized by its intricate structure. The lines connecting the nodes and forming loops suggest internal interactions and dependencies. The presence of the symbols "Y" and "ε" within the nodes likely represents specific variables or parameters within the system. The transition arrow "R1b" indicates a transformation or simplification process. The arrow's curvature suggests a non-linear or complex relationship between the initial and final states. The right state is a simple linear representation, indicating a reduction in complexity. The arrowheads at both ends suggest a continuous flow or a stable state. ### Key Observations The diagram highlights a clear transition from a complex, interconnected state to a simpler, linear state. The label "R1b" suggests a specific rule or operation governing this transition. The symbols "Y" and "ε" within the left state may represent key variables or parameters that are either eliminated or simplified during the transition. ### Interpretation This diagram likely represents a simplification or reduction of a complex system. The left state could represent a system with multiple interacting components, while the right state represents a simplified model or approximation. The transition "R1b" could represent a mathematical operation, a physical process, or a logical rule that transforms the complex state into a simpler one. The use of symbols "Y" and "ε" suggests a mathematical or scientific context. "ε" is often used to represent a small quantity or an error term, while "Y" could represent a variable or output. The diagram could be illustrating a process of approximation, where the complex system is reduced to a simpler model by neglecting certain terms or interactions. The diagram's simplicity suggests it is intended to convey a high-level understanding of the system's behavior, rather than a detailed technical specification. It could be used to illustrate a concept in a textbook, a presentation, or a research paper. The diagram is not providing any numerical data, but rather a conceptual representation of a system's state transition. </details> 2.4. (R2) move. This corresponds to the Reidemeister II move for emergent algebras. It involves an Υ graph and two other: a ¯ ε and a ¯ µ graph, with ε, µ ∈ Γ. <details> <summary>Image 15 Details</summary>  ### Visual Description \n ## Diagram: Rule Application R2 ### Overview The image depicts a diagram illustrating a rule application labeled "R2". It shows a transformation from a graph on the left to a graph on the right. Both graphs consist of nodes (circles) and directed edges (arrows). Numerical labels are associated with the edges. ### Components/Axes The diagram consists of two graphs connected by a bidirectional arrow labeled "R2". * **Left Graph:** Contains three nodes labeled "Y", "μ", and "ε". * **Right Graph:** Contains a single node labeled "εμ". * **Edges:** Each edge is labeled with a number: 1, 2, and 3. * **R2 Arrow:** A curved, bidirectional arrow labeled "R2" indicates the transformation between the two graphs. ### Detailed Analysis or Content Details **Left Graph:** * Node Y has two incoming edges: one labeled "1" and one from node ε. * Node μ has two incoming edges: one from node Y and one from node ε. * Node ε has two incoming edges: one labeled "2" and one from node Y. * Node μ has one outgoing edge labeled "3". **Right Graph:** * Node εμ has two incoming edges: one labeled "1" and one labeled "2". * Node εμ has one outgoing edge labeled "3". The transformation R2 appears to combine nodes Y and ε into a single node εμ, while preserving the edge labels. ### Key Observations The transformation R2 simplifies the graph by merging two nodes (Y and ε) into one (εμ). The edge labels are maintained during this transformation. The direction of the arrow suggests the transformation is reversible. ### Interpretation This diagram likely represents a simplification or reduction rule within a formal system, possibly related to graph rewriting or algebraic manipulation. The rule R2 appears to be a combination or merging operation. The preservation of edge labels suggests that the transformation is designed to maintain certain properties or relationships within the system. The bidirectional arrow indicates that the transformation is not necessarily a one-way process, and the original state can potentially be recovered from the transformed state. The labels 1, 2, and 3 likely represent some form of input or identifier associated with the edges. Without further context, it's difficult to determine the specific meaning of these labels or the overall purpose of the transformation. However, the diagram clearly illustrates a structural change governed by the rule R2. </details> This move appears in section 3.4, p. 21 [6], with the supplementary name 'triangle move'. - 2.5. (ext2) move. This corresponds to the rule (ext2) from λ -Scale calculus, it expresses the fact that in emergent algebras the operation indexed with the neutral element 1 of the group Γ has the property x ◦ 1 y = y . <details> <summary>Image 16 Details</summary>  ### Visual Description \n ## Diagram: Phylogenetic Tree Representation ### Overview The image depicts a diagram resembling a phylogenetic tree, illustrating a branching evolutionary relationship. It shows two tree structures connected by an arc labeled "ext 2". Each branch is labeled with a numerical value, likely representing a species or group. ### Components/Axes The diagram consists of: * **Nodes:** Represented by circles and branch endpoints. * **Branches:** Lines connecting the nodes, indicating evolutionary relationships. * **Labels:** Numerical values (1, 2, 3) associated with each branch endpoint. * **Arc:** A curved arrow labeled "ext 2" connecting the two tree structures. There are no explicit axes in this diagram. ### Detailed Analysis or Content Details The diagram can be divided into two main tree structures. **Left Tree:** * A central node labeled "1" is the root of the tree. * Three branches originate from this central node. * The top branch is labeled "3". * The bottom-left branch is labeled "1". * The bottom-right branch is labeled "2". **Right Tree:** * A branch originates from an implied root. * The top branch is labeled "3". * The bottom-left branch is labeled "1". * The bottom-right branch is labeled "2". The arc labeled "ext 2" originates near the central node "1" of the left tree and points towards the implied root of the right tree. ### Key Observations * Both trees share the same branch labels (1, 2, 3), suggesting a common ancestry or a relationship between the groups represented. * The arc "ext 2" implies an extinction event or a significant evolutionary divergence. * The trees are not identical in structure, indicating that the evolutionary paths diverged after the event represented by "ext 2". ### Interpretation The diagram likely represents the evolutionary history of a group of organisms. The central node "1" could represent a common ancestor. The branching pattern shows how this ancestor gave rise to different lineages, labeled 1, 2, and 3. The arc labeled "ext 2" suggests that one lineage experienced an extinction event or a major evolutionary shift, leading to the divergence of the two trees. The fact that the same labels (1, 2, 3) appear in both trees suggests that the surviving lineages retained some characteristics of the original ancestor. The diagram is a simplified representation of a complex evolutionary process. It does not provide information about the time scale or the specific characteristics of the organisms involved. However, it effectively illustrates the concept of common ancestry and evolutionary divergence. The "ext 2" label is crucial, indicating a significant event that shaped the evolutionary trajectory of the group. </details> 2.6. Local pruning. Local pruning moves are local moves which eliminate 'dead' edges. Notice that, unlike the previous moves, these are one-way (you can eliminate dead edges, but not add them to graphs). <details> <summary>Image 17 Details</summary>  ### Visual Description \n ## Diagram: LOC Pruning Illustration ### Overview The image depicts a diagram illustrating the concept of "LOC Pruning". It shows a transformation from a complex node structure to a simplified linear structure through a pruning process. The diagram consists of two sets of transformations, each with an initial node, a pruning arrow, and a resulting linear structure. ### Components/Axes The diagram contains the following components: * **Nodes:** Represented by circles with internal symbols (λ and a Y-shaped symbol). * **Arrows:** Indicate input and output connections to the nodes. * **Pruning Arrows:** Blue arrows labeled "LOC PRUNING" showing the transformation direction. * **Linear Structures:** Vertical lines with numerical labels (1 and 2) indicating input/output points. * **Labels:** "λ" inside the first node, and "LOC PRUNING" labeling the pruning arrows. * **Numerical Labels:** "1" and "2" on the vertical lines. ### Detailed Analysis or Content Details **Transformation 1 (Top):** * **Initial Node:** A circle containing the symbol "λ". Three arrows point towards the circle, and three arrows point away from it. * **Pruning Arrow:** A blue arrow labeled "LOC PRUNING" points from the initial node to the right. * **Resulting Structure:** A single vertical line with an arrow pointing upwards. **Transformation 2 (Bottom):** * **Initial Node:** A circle containing a Y-shaped symbol. Two arrows point towards the circle, and two arrows point away from it. * **Pruning Arrow:** A blue arrow labeled "LOC PRUNING" points from the initial node to the left. * **Resulting Structure:** A single vertical line with an arrow pointing upwards. The line is labeled with "1" at the bottom and "2" at the top. ### Key Observations * The "LOC Pruning" process appears to simplify a complex node with multiple connections into a single linear pathway. * The pruning arrows indicate a directional transformation. * The numerical labels (1 and 2) on the bottom structure suggest a sequential flow or ordering. * The initial nodes have different internal symbols (λ and Y-shaped), suggesting different types of initial structures. ### Interpretation The diagram illustrates a simplification process called "LOC Pruning". The initial nodes represent complex operations or decision points with multiple inputs and outputs. The "LOC Pruning" process reduces this complexity to a single linear pathway, potentially representing a streamlined or optimized operation. The numerical labels on the bottom structure suggest a sequential order of operations or data flow. The different symbols within the initial nodes suggest that the pruning process can be applied to different types of complex structures. The diagram doesn't provide quantitative data, but rather a conceptual illustration of a transformation. It suggests that LOC Pruning is a method for reducing complexity and streamlining processes. The use of arrows and labels clearly communicates the direction and nature of the transformation. The diagram is a visual representation of an algorithmic or computational process. </details> <details> <summary>Image 18 Details</summary>  ### Visual Description \n ## Diagram: Loc Pruning Illustration ### Overview The image presents a series of diagrams illustrating a process labeled "LOC PRUNING" applied to tree-like structures. The diagrams show a tree being simplified by removing branches. There are three distinct stages depicted, each showing a tree structure and its transformation. ### Components/Axes The diagrams consist of: * **Tree Structures:** Represented by branching lines, with a central node labeled "ε" (epsilon) in the first two diagrams. * **Branches:** Labeled with the numbers "1" and "2". * **Arrows:** Curved blue arrows indicating the "LOC PRUNING" operation. * **Text Labels:** "LOC PRUNING" appears above each arrow. * **Red Dashed Circles:** Enclosing simplified tree structures in the third diagram. ### Detailed Analysis or Content Details **Diagram 1 (Top):** * A tree with a central node "ε" and two branches labeled "1" and "2". * A curved blue arrow originates from the central node and points towards the right, labeled "LOC PRUNING". * To the right of the arrow are two simplified branches, both pointing upwards. The branch corresponding to "1" is slightly shorter than the branch corresponding to "2". **Diagram 2 (Middle):** * Identical to Diagram 1 in its initial tree structure. * A curved blue arrow originates from the central node and points towards the right, labeled "LOC PRUNING". * To the right of the arrow are two simplified branches, both pointing upwards. The branch corresponding to "1" is slightly shorter than the branch corresponding to "2". **Diagram 3 (Bottom):** * A simplified tree structure enclosed within a red dashed circle. The tree consists of a single upward-pointing branch. * A curved blue arrow originates from the simplified tree and points towards the right, labeled "LOC PRUNING". * To the right of the arrow is another simplified tree structure, also enclosed within a red dashed circle, consisting of a single upward-pointing branch. ### Key Observations * The "LOC PRUNING" operation consistently simplifies the tree structure by removing branches. * The initial tree structures in the first two diagrams are identical. * The final structures in the third diagram are identical. * The epsilon symbol "ε" is only present in the first two diagrams. * The branches are labeled with numerical identifiers "1" and "2". ### Interpretation The diagrams illustrate a pruning process, likely within a larger algorithm or data structure. "LOC PRUNING" appears to be a simplification step, reducing a tree-like structure to its essential components. The epsilon symbol "ε" might represent a tolerance or threshold value used in the pruning process. The consistent application of "LOC PRUNING" suggests a deterministic operation. The numerical labels "1" and "2" on the branches could represent priorities or weights associated with each branch, influencing the pruning decision. The red dashed circles in the final diagram emphasize the resulting simplified structures after pruning. The diagrams demonstrate a reduction in complexity, potentially for optimization or efficiency purposes. The diagrams do not provide any quantitative data, but rather a visual representation of a process. </details> Global moves or conditions. Global moves are those which are not local, either because the condition C applies to parts of the graph which may have an arbitrary large sum or edges plus nodes, or because after the move the graph P ′ which replaces the graph P has an arbitrary large sum or edges plus nodes. 2.7. (ext1) move. This corresponds to the rule (ext1) from λ -Scale calculus, or to η -reduction in lambda calculus (see theorem 3.1, part (e) for details). It involves a λ graph (think about the λ abstraction operation in lambda calculus) and a graph (think about the application operation in lambda calculus). The rule is: if there is no oriented path from '2' to '1', then the following move can be performed. <details> <summary>Image 19 Details</summary>  ### Visual Description \n ## Diagram: System Flow with External Interaction ### Overview The image depicts a diagram representing a system with two internal components and an external interaction. The system consists of a component labeled "λ" and another component represented by a three-way branching symbol. Arrows indicate the flow of information or interaction between these components and with an external entity labeled "ext 1". Numerical labels "1" and "2" are present, likely indicating input or output points. ### Components/Axes * **Components:** * λ (Lambda) - Represented by a circle. * Three-way branching symbol - Represented by a circle with three lines emanating from it. * ext 1 - Represents an external entity. * **Labels:** * "λ" - Inside the first circular component. * "ext 1" - Above the bidirectional arrow. * "1" - Below the circular component labeled "λ". * "2" - Above and to the left of the circular component labeled "λ". * **Arrows:** * Arrows indicate the direction of flow or interaction. * A bidirectional arrow connects "ext 1" to the system. * Arrows connect the "λ" component to the three-way branching component and back. * An arrow originates from the three-way branching component and points towards the "λ" component. ### Detailed Analysis or Content Details The diagram shows a closed loop between the "λ" component and the three-way branching component. The flow starts at "1", enters the "λ" component, then flows to the three-way branching component, and returns to "λ". There is also an external interaction labeled "ext 1" which interacts bidirectionally with the system. * **Flow 1:** From "1" to "λ". * **Flow 2:** From "λ" to the three-way branching component. * **Flow 3:** From the three-way branching component back to "λ". * **External Interaction:** Bidirectional arrow between "ext 1" and the system (likely interacting with the "λ" component). * **Input/Output:** "1" and "2" likely represent input or output points for the system. ### Key Observations The diagram illustrates a feedback loop within the system, with an external interaction influencing or being influenced by the internal processes. The "λ" component appears central to the system's operation. The three-way branching component suggests a decision-making or distribution point within the loop. ### Interpretation This diagram likely represents a control system or a process with feedback. The "λ" component could be a controller or a core processing unit. The three-way branching component could represent a switch, a router, or a point where different actions are taken based on the system's state. The external interaction "ext 1" suggests that the system is not isolated and interacts with its environment. The numerical labels "1" and "2" could represent input and output signals or states. The closed loop indicates that the system's output influences its input, creating a self-regulating or iterative process. Without further context, it's difficult to determine the specific function of the system, but the diagram clearly illustrates a dynamic interaction between internal components and an external environment. The diagram is a conceptual representation and does not provide quantitative data. </details> 2.8. (Global FAN-OUT) move. This is a global move, because it consists in replacing (under certain circumstances) a graph by two copies of that graph. The rule is: if a graph in G ∈ GRAPH has a Υ bottleneck, that is if we can find a sub-graph A ∈ GRAPH connected to the rest of the graph G only through a Υ gate, then we can perform the move explained in the next figure, from the left to the right. Conversely, if in the graph G we can find two identical subgraphs (denoted by A ), which are in GRAPH , which have no edge connecting one with another and which are connected to the rest of G only through one edge, as in the RHS of the figure, then we can perform the move from the right to the left. <details> <summary>Image 20 Details</summary>  ### Visual Description \n ## Diagram: Global Fan-Out Illustration ### Overview The image presents a diagram illustrating a concept labeled "GLOBAL FAN-OUT". It depicts two circular arrangements of nodes and connections, visually representing a transformation or distribution process. The left side shows a central node with multiple outgoing connections, while the right side shows a distribution of those connections to separate nodes. ### Components/Axes The diagram consists of the following elements: * **Circles:** Two dashed red circles, one on the left and one on the right, enclosing nodes and connections. * **Nodes:** Represented by circles. The left side has a central node with a smaller, split node below it. The right side has two separate nodes. * **Connections/Arrows:** Arrows indicating the flow or relationship between nodes. * **Labels:** * "1" - Labeling the top-left connection/node on both sides. * "2" - Labeling the top-right connection/node on both sides. * "A" - Labeling the bottom connection/node on both sides. * **Text:** "GLOBAL FAN-OUT" - positioned below the two circular arrangements, with a curved blue arrow pointing from the left to the right. ### Detailed Analysis or Content Details The diagram shows a transformation from a single point with multiple outputs to multiple points, each receiving a portion of the original outputs. **Left Side:** * A central node has three outgoing connections labeled 1, 2, and A. * The central node has a smaller, split node below it, with a Y-shape. * The connections 1, 2, and A originate from the central node. **Right Side:** * Two separate nodes are present. * Connection 1 leads to the left node, labeled "A". * Connection 2 leads to the right node, labeled "A". * The connections 1 and 2 originate from outside the diagram, implied by the arrow. The curved blue arrow indicates a directional flow from the left side to the right side. ### Key Observations * The diagram visually represents a distribution or fan-out process. * The label "GLOBAL FAN-OUT" suggests a broad or system-wide distribution. * The "A" label appears on both sides, indicating a consistent element in the transformation. * The split node on the left side suggests a branching or decision point before the fan-out occurs. ### Interpretation The diagram illustrates the concept of "GLOBAL FAN-OUT," likely in a computing or networking context. It demonstrates how a single source (the central node on the left) can distribute its output to multiple destinations (the two nodes on the right). The split node on the left could represent a load balancer or a routing mechanism that determines how the output is distributed. The "A" label might represent a specific type of data or a common destination. The curved arrow emphasizes the direction of the distribution process. The diagram is a simplified representation of a complex system, focusing on the core concept of distributing a signal or data stream. It doesn't provide specific numerical data or performance metrics, but rather a conceptual overview of the fan-out process. The diagram suggests a transformation where a centralized process is expanded into a distributed one. </details> Remark that (global FAN-OUT) trivially implies (CO-COMM). ( As an local rule alternative to the global FAN-OUT, we might consider the following. Fix a number N and consider only graphs A which have at most N (nodes + arrows). The N LOCAL FAN-OUT move is the same as the GLOBAL FAN-OUT move, only it applies only to such graphs A . This local FAN-OUT move does not imply CO-COMM.) 2.9. Global pruning. This a global move which eliminates 'dead' edges. The rule is: if a graph in G ∈ GRAPH has a ending, that is if we can find a sub-graph A ∈ GRAPH connected only to a gate, with no edges connecting to the rest of G , then we can erase this graph and the respective gate. <details> <summary>Image 21 Details</summary>  ### Visual Description \n ## Diagram: Global Pruning ### Overview The image depicts a diagram illustrating a process labeled "GLOBAL PRUNING". It shows a circular region containing a symbol, with an arrow indicating a transfer or removal of content to another empty circular region. ### Components/Axes The diagram consists of: * Two circular regions, both outlined with dashed red lines. * A symbol within the left circular region, resembling a horizontal line with a vertical line extending upwards from its center. This symbol is labeled "A". * An arrow originating from the center of the symbol "A" and pointing towards the center of the right circular region. * Text "GLOBAL PRUNING" positioned below the arrow. ### Detailed Analysis or Content Details The left circular region contains the symbol "A". The symbol appears to be a representation of a variable or a measurement, with the vertical line indicating a value or direction. The arrow indicates a transfer or removal of the content represented by "A" from the left circle to the right circle. The text "GLOBAL PRUNING" suggests that this transfer or removal is part of a larger process of reducing or eliminating elements. ### Key Observations The diagram is simple and conceptual. It doesn't contain numerical data or specific measurements. The focus is on the process of "GLOBAL PRUNING" and the movement of an element "A" as part of that process. The empty right circle suggests a destination or a state after the pruning process. ### Interpretation The diagram likely represents a concept in machine learning, optimization, or data processing. "GLOBAL PRUNING" suggests a technique for reducing the complexity of a model or dataset by removing less important elements. The symbol "A" could represent a feature, a connection, or a data point that is being pruned. The arrow indicates that this element is being removed from its original context and potentially discarded or moved to a different location. The diagram highlights the idea of selectively removing elements to improve efficiency or performance. The lack of specific details suggests that this is a high-level illustration of a general principle rather than a specific implementation. </details> The global pruning may be needed because of the λ gates, which cannot be removed only by local pruning. 2.10. Elimination of loops. It is possible that, after using a local or global move, we obtain a graph with an arrow which closes itself, without being connected to any node. Here is an example, concerning the application of the graphic β move. We may erase any such loop, or add one. λ GRAPHS. The edges of an elementary graph λ can be numbered unambiguously, clockwise, by 1, 2, 3, such that 1 is the number of the entrant edge. Definition 2.4 A graph G ∈ GRAPH is a λ -graph, notation G ∈ λGRAPH , if: - -it does not have ¯ ε gates, - -for any node λ any oriented path in G starting at the edge 2 of this node can be completed to a path which either terminates in a graph , or else terminates at the edge 1 of this node. The condition G ∈ λGRAPH is global, in the sense that in order to decide if G ∈ λGRAPH we have to examine parts of the graph which may have an arbitrary large sum or edges plus nodes. ## 3 Conversion of lambda terms into GRAPH Here I show how to associate to a lambda term a graph in GRAPH , then I use this to show that β -reduction in lambda calculus transforms into the β rule for GRAPH . (Thanks to Morita Yasuaki for some corrections.) Indeed, to any term A ∈ T ( X ) (where T ( X ) is the set of lambda terms over the variable set X ) we associate its syntactic tree. The syntactic tree of any lambda term is constructed by using two gates, one corresponding to the λ abstraction and the other corresponding to the application. We draw syntactic trees with the leaves (elements of X ) at the bottom and the root at the top. We shall use the following notation for the two gates: at the left is the gate for the λ abstraction and at the right is the gate for the application. <details> <summary>Image 22 Details</summary>  ### Visual Description \n ## Diagram: Lambda Calculus Tree Representations ### Overview The image presents two tree-like diagrams representing expressions in lambda calculus. Both diagrams utilize a branching structure to visually depict the application of functions to arguments. The diagrams are positioned side-by-side for comparison. ### Components/Axes The diagrams consist of nodes (circles) and branches connecting them. Each branch is labeled with a variable or expression. The diagrams do not have explicit axes or legends in the traditional sense of a chart. The labels are directly embedded within the diagram structure. ### Detailed Analysis or Content Details **Diagram 1 (Left):** * **Root Node:** A circle containing the lambda symbol "λ". * **Branches from Root:** Two branches originate from the root node. * The left branch is labeled "x". * The right branch is labeled "A". * **Node above Root:** A node above the root node is labeled "λx.A". * **Overall Structure:** The diagram represents the lambda abstraction "λx.A", where 'x' is the bound variable and 'A' is the body of the lambda expression. **Diagram 2 (Right):** * **Root Node:** A circle containing the lambda symbol "λ". * **Branches from Root:** Two branches originate from the root node. * The left branch is labeled "A". * The right branch is labeled "B". * **Node above Root:** A node above the root node is labeled "AB". * **Overall Structure:** The diagram represents the application of a lambda abstraction to arguments 'A' and 'B', resulting in the expression "AB". ### Key Observations Both diagrams use the same visual structure to represent lambda calculus expressions. The lambda symbol (λ) consistently appears at the root of each tree. The branching structure clearly shows the application of functions to arguments. The diagrams are simple and focus on the core structure of lambda expressions. ### Interpretation The diagrams illustrate the graphical representation of lambda calculus expressions. The first diagram shows a lambda abstraction, defining a function that takes an argument 'x' and returns 'A'. The second diagram shows the application of this function (or a similar one) to arguments 'A' and 'B', resulting in the expression 'AB'. The diagrams are useful for visualizing the structure of lambda expressions and understanding how functions are applied to arguments. They demonstrate a way to represent abstract mathematical concepts in a visual format. The diagrams are not providing data, but rather a visual representation of a formal system. They are meant to aid in understanding the syntax and semantics of lambda calculus. </details> Remark that these two gates are from the graphical alphabet of GRAPH , but the syntactic tree is decorated: at the bottom we have leaves from X . Also, remark the peculiar orientation of the edge from the left (in tree notation convention) of the λ gate. For the moment, this orientation is in contradiction with the implicit orientation (from down-up) of edges of the syntactic tree, but soon this matter will become clear. We shall remove all leaves decorations, with the price of introducing new gates, namely Υ and gates. This will be done in a sequence of steps, detailed further. Take the syntactic tree of A ∈ T ( X ), drawn with the mentioned conventions (concerning gates and the positioning of leaves and root respectively). We take as examples the following five lambda terms: I = λx.x , K = λx. ( λy.x ), S = λx. ( λy. ( λz. (( xz )( yz )))), Ω = ( λx. ( xx ))( λx. ( xx )) and T = ( λx. ( xy ))( λx. ( xy )). Step 1. Elimination of bound variables, part I. Any leaf of the tree is connected to the root by an unique path. Start from the leftmost leaf, perform the algorithm explained further, then go to the right and repeat until all leaves are exhausted. We initialize also a list B = ∅ of bound variables. Take a leaf, say decorated with x ∈ X . To this leaf is associated a word (a list) which is formed by the symbols of gates which are on the path which connects (from the bottom-up) the leaf with the root, together with information about which way, left (L) or right (R), the path passes through the gates. Such a word is formed by the letters λ L , λ R , L , R . If the first letter is λ L then add to the list B the pair ( x, w ( x )) formed by the variable name x , and the associated word (describing the path to follow from the respective leaf to the root). Then pass to a new leaf. Else continue along the path to the roof. If we arrive at a λ gate, this can happen only coming from the right leg of the λ gate, thus we can find only the letter λ R . In such a case look at the variable y which decorates the left leg of the same λ gate. If x = y then add to the syntactic tree a new edge, from y to x and proceed further along the path, else proceed further. If the root is attained then pass to next leaf. Examples: the graphs associated to the mentioned lambda terms, together with the list of bound variables, are the following. $$\begin{aligned} - I &= x _ { x } . x _ { y } \text { has } B = \{ ( x , y ) \mid ( x z ) ( y z ) \} \\ &= \left\{ \begin{array}{l} x _ { x } ^ { L } \cdot y _ { y } ^ { L } \times R \right\} , S = \left\{ \begin{array}{l} x _ { R } ^ { L } \cdot y _ { R } ^ { L } \times R \right\} . \end{aligned}$$ <details> <summary>Image 23 Details</summary>  ### Visual Description \n ## Diagram: Lambda Calculus Reduction Examples ### Overview The image presents three diagrams illustrating lambda calculus reduction steps. Each diagram depicts a lambda expression and its reduction, represented as a tree-like structure with labeled nodes and arrows indicating the reduction process. The diagrams demonstrate how lambda expressions are simplified by applying beta reduction. ### Components/Axes The diagrams consist of: * **Nodes:** Represented by circles, labeled with either "λ" (lambda) or variables (x, y, z). * **Arrows:** Red curved arrows indicate the reduction process, showing how a lambda abstraction is applied to an argument. * **Input/Output:** Each diagram shows an initial lambda expression and its reduced form. ### Detailed Analysis or Content Details **Diagram 1 (Leftmost):** * The initial expression is a lambda abstraction with a single variable 'x' and a self-application. * The lambda node (λ) has two outgoing edges, both labeled 'x', and these edges loop back to the lambda node itself. * The reduction arrow starts from the bottom-left 'x' and curves upwards to the lambda node. **Diagram 2 (Center):** * The initial expression is a lambda abstraction with two variables 'x' and 'y'. * The lambda node (λ) has two branches. The first branch leads to 'x', and the second branch splits into two further branches, labeled 'y' and 'x'. * The reduction arrow starts from the bottom 'x' and curves upwards to the lambda node. **Diagram 3 (Rightmost):** * This diagram shows a more complex reduction. * There are two lambda nodes (λ). * The first lambda node has outgoing edges to 'x' and 'y'. * The second lambda node has a more complex structure, branching to 'z', and then further branching to 'x', 'z', 'y', and 'z'. * Multiple reduction arrows are present, indicating multiple reduction steps. The arrows originate from 'x', 'y', 'z', and the lower branches of the second lambda node, and curve upwards towards the lambda nodes. ### Key Observations * The diagrams demonstrate the application of beta reduction, where a lambda abstraction is applied to an argument. * The complexity of the diagrams increases from left to right, showing more intricate reduction steps. * The use of red arrows consistently indicates the direction of reduction. * The diagrams do not contain numerical data, but rather illustrate a conceptual process. ### Interpretation The diagrams visually represent the process of lambda calculus reduction. They demonstrate how complex lambda expressions can be simplified by repeatedly applying beta reduction. The increasing complexity of the diagrams suggests that the reduction process can become more involved as the initial expression becomes more intricate. The diagrams are a pedagogical tool for understanding the core concepts of lambda calculus, a foundational model of computation. The diagrams show how a function (represented by the lambda abstraction) is applied to an argument (represented by the variables x, y, and z), resulting in a simplified expression. The arrows indicate the substitution process, where the argument replaces the formal parameter in the function body. The diagrams are not providing specific data, but rather illustrating a process. They are a visual representation of a formal system. </details> $$\begin{aligned} - \Omega &= ( x _ { x } ( x _ { y } ) ) ( x _ { x } ( x _ { y } ) \\ &= \{ ( x _ { x } , λ ^ { L } , L ) , ( x _ { x } , λ ^ { L } , R ) \} , T = \{ L , R \} , \\ \lambda ^ { L } &\text{has } B = \{ ( x _ { x } , λ ^ { L } , L ) , ( x _ { x } , λ ^ { L } , R ) \}, T = \{ L , R \} . \end{aligned}$$ Step 2. Elimination of bound variables, part II. We have now a list B of bound variables. If the list is empty then go to the next step. Else, do the following, starting from the first element of the list, until the list is finished. <details> <summary>Image 24 Details</summary>  ### Visual Description \n ## Diagram: Lambda Calculus Reduction ### Overview The image presents two diagrams illustrating a reduction process in lambda calculus. Both diagrams depict tree-like structures representing lambda expressions, with arrows indicating the reduction steps. The diagrams are positioned side-by-side for comparison. ### Components/Axes The diagrams consist of nodes connected by branches. Each node contains either the lambda symbol (λ), a variable (x or y), or a branching symbol (a 'Y' shape indicating function application). Red curved arrows indicate the reduction steps, pointing from the node being reduced to its replacement. ### Detailed Analysis or Content Details **Diagram 1 (Left)** * **Top Node:** A branching node (Y-shape) with an upward-pointing arrow. * **Second Level:** Two nodes branching from the top node, both labeled with the lambda symbol (λ). * **Third Level (Left Branch):** A branching node (Y-shape) with an upward-pointing arrow, labeled 'x' on the incoming branch. * **Fourth Level (Left Branch):** Two nodes branching from the third level node, both labeled 'x'. * **Third Level (Right Branch):** A branching node (Y-shape) with an upward-pointing arrow, labeled 'x' on the incoming branch. * **Fourth Level (Right Branch):** Two nodes branching from the third level node, both labeled 'x'. * **Reduction Arrows:** Three red curved arrows. * The first arrow starts at the bottom-left 'x' node and curves upwards to the third level node labeled 'x'. * The second arrow starts at the bottom-right 'x' node and curves upwards to the third level node labeled 'x'. * The third arrow starts at the third level node labeled 'x' (left branch) and curves upwards to the top node. **Diagram 2 (Right)** * **Top Node:** A branching node (Y-shape) with an upward-pointing arrow. * **Second Level:** Two nodes branching from the top node, both labeled with the lambda symbol (λ). * **Third Level (Left Branch):** A branching node (Y-shape) with an upward-pointing arrow, labeled 'x' on the incoming branch. * **Fourth Level (Left Branch):** Two nodes branching from the third level node, one labeled 'x' and the other 'y'. * **Third Level (Right Branch):** A branching node (Y-shape) with an upward-pointing arrow, labeled 'x' on the incoming branch. * **Fourth Level (Right Branch):** Two nodes branching from the third level node, one labeled 'x' and the other 'y'. * **Reduction Arrows:** Two red curved arrows. * The first arrow starts at the bottom-left 'x' node and curves upwards to the third level node labeled 'x'. * The second arrow starts at the bottom-right 'x' node and curves upwards to the third level node labeled 'x'. ### Key Observations The diagrams demonstrate a reduction process where variables are substituted into lambda expressions. Diagram 1 shows a case where all variables are 'x', while Diagram 2 shows a case where the variables are 'x' and 'y'. The arrows indicate the substitution steps. ### Interpretation These diagrams likely illustrate beta-reduction, a fundamental operation in lambda calculus. Beta-reduction involves substituting a variable in a lambda expression with another expression. The diagrams show how this substitution unfolds within the tree structure of the lambda expression. The difference between the two diagrams highlights how the presence of different variables ('x' and 'y' in Diagram 2) affects the reduction process. The diagrams are a visual representation of the application of a function to its arguments, and the subsequent simplification of the expression. The diagrams are not providing numerical data, but rather a visual representation of a logical process. </details> An element, say ( x, w ( x )), of the list, is either connected to other leaves by one or more edges added at step 1, or not. If is not connected then erase the variable name with the associated path w ( x ) and replace it by a gate. If it is connected then erase it, replace it by a tree formed by Υ gates, which starts at the place where the element of the list were before the erasure and stops at the leaves which were connected to x . Erase all decorations which were joined to x and also erase all edges which were added at step 1 to the leave x from the list. Examples: after the step 2, the graphs associated to the mentioned lambda terms are the following. - -the graphs of I = λx.x , K = λx. ( λy.x ), S = λx. ( λy. ( λz. (( xz )( yz )))) are - -the graphs of Ω = ( λx. ( xx ))( λx. ( xx )), T = ( λx. ( xy ))( λx. ( xy )) are <details> <summary>Image 25 Details</summary>  ### Visual Description \n ## Diagram: Lambda Calculus Reduction Steps ### Overview The image presents a series of diagrams illustrating the reduction steps in lambda calculus. It shows how a lambda expression is simplified through beta reduction, starting with a simple self-application and progressing to a more complex nested structure. The diagrams are arranged horizontally, representing sequential steps in the reduction process. ### Components/Axes The diagrams consist of circular and Y-shaped nodes connected by arrows. Each node contains the symbol "λ" (lambda). The arrows indicate the direction of reduction or application. There are no explicit axes or scales. The diagrams are purely structural, representing the transformation of lambda expressions. ### Detailed Analysis or Content Details **Diagram 1 (Leftmost):** * A single circular node labeled "λ". * An arrow originates from the top of the node and curves back down to the node itself, indicating self-application. * A curved arrow originates from the bottom of the node and curves back up to the node itself. **Diagram 2 (Middle):** * A circular node labeled "λ". * An arrow originates from the top of the node and points downwards. * A second circular node labeled "λ" is positioned below the first. * An arrow originates from the second node and points upwards to the first node. * A small vertical line is present below the second node. **Diagram 3 (Rightmost):** * Three circular nodes labeled "λ" are positioned horizontally. * The leftmost "λ" node has an arrow pointing to the middle "λ" node. * The middle "λ" node has an arrow pointing back to the leftmost "λ" node. * The middle "λ" node also has two arrows pointing downwards. * Each of these downward arrows connects to a Y-shaped node. * Each Y-shaped node has three branches, with arrows pointing from the central point of the Y to each branch. * The bottom-most Y-shaped node has arrows pointing upwards to the two Y-shaped nodes above it. ### Key Observations The diagrams demonstrate a progression from a simple lambda expression to a more complex, nested structure. The introduction of the Y-shaped nodes suggests the creation of a fixed-point combinator, likely representing the Y combinator. The diagrams show how self-application and recursion are represented in lambda calculus. ### Interpretation The image illustrates the process of beta reduction in lambda calculus, specifically demonstrating how a lambda expression can be reduced to its normal form. The first diagram represents a simple lambda abstraction. The second diagram shows a basic application. The third diagram demonstrates the construction of a recursive function using the Y combinator. The Y combinator allows for the definition of recursive functions in a purely functional setting without explicit recursion. The diagrams are a visual representation of the mathematical operations involved in lambda calculus, highlighting the power and elegance of this formal system. The diagrams are not providing numerical data, but rather a structural representation of a computational process. The diagrams are a visual aid for understanding the core concepts of lambda calculus and functional programming. </details> <details> <summary>Image 26 Details</summary>  ### Visual Description \n ## Diagram: Lambda Calculus Representations ### Overview The image presents two diagrams representing lambda calculus expressions. Both diagrams utilize a tree-like structure with nodes labeled with either 'λ' (lambda) or unlabeled circles representing variables. Arrows indicate the flow of substitution or application. The diagrams visually depict the structure of lambda terms. ### Components/Axes The diagrams consist of nodes (circles) and directed edges (arrows). The nodes are labeled with 'λ', 'y', or are left unlabeled, representing variables. The arrows show the direction of function application or variable substitution. There are no explicit axes or scales. ### Detailed Analysis or Content Details **Diagram 1 (Left):** * **Top Node:** An unlabeled circle with an outgoing arrow pointing upwards. * **Second Layer:** Two nodes labeled 'λ', branching from the top node. * **Third Layer (Left Branch):** Two unlabeled circles connected by a bidirectional arrow, and an arrow from the first circle to the 'λ' node above. * **Third Layer (Right Branch):** Two unlabeled circles connected by a bidirectional arrow, and an arrow from the first circle to the 'λ' node above. * **Overall Structure:** The diagram represents a lambda expression with two branches, each containing a lambda abstraction and a cycle between two variables. **Diagram 2 (Right):** * **Top Node:** An unlabeled circle with an outgoing arrow pointing upwards. * **Second Layer:** Two nodes labeled 'λ', branching from the top node. * **Third Layer (Left Branch):** An unlabeled circle with a loop back to the 'λ' node above, and an arrow pointing downwards labeled 'y'. * **Third Layer (Right Branch):** An unlabeled circle with a loop back to the 'λ' node above, and an arrow pointing downwards labeled 'y'. * **Overall Structure:** The diagram represents a lambda expression with two branches, each containing a lambda abstraction and a self-looping variable, with a 'y' variable as output. ### Key Observations * Both diagrams share a similar top-level structure: an unlabeled node branching into two lambda abstractions. * The left diagram features cycles between variables within each lambda branch, while the right diagram uses self-loops and a 'y' variable. * The diagrams are visually distinct, representing different lambda calculus expressions. ### Interpretation The diagrams illustrate different lambda calculus terms. The left diagram likely represents a more complex expression involving mutual recursion or a fixed-point computation due to the cycles. The right diagram represents a simpler expression, potentially a lambda abstraction that returns the 'y' variable. The diagrams demonstrate how lambda calculus can be visually represented using a graph-like structure, where nodes represent functions or variables, and edges represent application or substitution. The absence of numerical data means we can only interpret the structural relationships between the components. The diagrams are not providing quantitative data, but rather a qualitative representation of lambda calculus expressions. The 'y' in the right diagram suggests a potential output or argument to the lambda function. The diagrams are likely used to explain or visualize the concepts of lambda calculus, such as abstraction, application, and recursion. </details> Remark that at this step the necessity of having the peculiar orientation of the left leg of the λ gate becomes clear. Remark also that there may be more than one possible tree of gates Υ, at each elimination of a bound variable (in case a bound variable has at least tree occurrences). One may use any tree of Υ which is fit. The problem of multiple possibilities is the reason of introducing the (CO-ASSOC) move. Step 3. We may still have leaves decorated by free variables. Starting from the left to the right, group them together in case some of them occur in multiple places, then replace the multiple occurrences of a free variable by a tree of Υ gates with a free root, which ends exactly where the occurrences of the respective variable are. Again, there are multiple ways of doing this, but we may pass from one to another by a sequence of (CO-ASSOC) moves. Examples: after the step 3, all the graphs associated to the mentioned lambda terms, excepting the last one, are left unchanged. The graph of the last term, changes. - -as an illustration, I figure the graphs of Ω = ( λx. ( xx ))( λx. ( xx )), left unchanged by step 3, and the graph of T = ( λx. ( xy ))( λx. ( xy )): <details> <summary>Image 27 Details</summary>  ### Visual Description \n ## Diagram: Lambda Calculus Reduction Diagrams ### Overview The image presents two diagrams illustrating the reduction process in lambda calculus. Both diagrams depict a series of nodes connected by directed edges, representing the application of lambda reduction rules. The diagrams show how a complex expression is simplified through successive substitutions. ### Components/Axes The diagrams consist of the following components: * **Nodes:** Represented by circles. Some nodes are labeled with the Greek letter "λ" (lambda), indicating a lambda abstraction. Other nodes are unlabeled, representing variables or application results. * **Directed Edges:** Arrows indicating the direction of reduction or substitution. * **Arrows pointing upwards:** Indicate the start and end points of the reduction process. ### Detailed Analysis or Content Details **Diagram 1 (Left):** * The diagram starts with a single node at the top, with an upward-pointing arrow. * This node branches into two identical sub-diagrams. * Each sub-diagram contains a node labeled "λ" at the center. * The "λ" node connects to two unlabeled nodes at the bottom, forming a triangle. * Each of the bottom nodes has a self-loop (an arrow pointing back to itself) and an arrow connecting it to the other bottom node. * The "λ" node also has an arrow pointing upwards to the top node. **Diagram 2 (Right):** * The diagram starts with a single node at the top, with an upward-pointing arrow. * This node branches into two nodes labeled "λ". * Each "λ" node connects to an unlabeled node. * Each unlabeled node has a self-loop and an arrow connecting it back to the "λ" node it originated from. * The bottom two unlabeled nodes are connected by an arrow, forming a loop. * The bottom node also has an upward-pointing arrow. ### Key Observations * Both diagrams represent a reduction process, but the structure of the reduction differs. * Diagram 1 shows a parallel reduction of two identical expressions. * Diagram 2 shows a sequential reduction, with the result of one reduction feeding into the next. * The self-loops in both diagrams indicate repeated application of a reduction rule. * The diagrams are purely structural and do not contain numerical data. ### Interpretation The diagrams illustrate different strategies for evaluating lambda expressions. Diagram 1 represents a parallel reduction strategy, where both branches of the expression are reduced simultaneously. Diagram 2 represents a sequential reduction strategy, where one branch is reduced before the other. The choice of reduction strategy can affect the efficiency and termination of the evaluation process. The diagrams are abstract representations of the reduction process and do not specify the exact lambda expression being reduced. They are intended to convey the general structure of the reduction process rather than the specific details of a particular expression. The diagrams are useful for understanding the concepts of lambda calculus and the different ways in which lambda expressions can be evaluated. The diagrams are not providing facts or data, but rather a visual representation of a process. </details> Theorem 3.1 Let A ↦→ [ A ] be a transformation of a lambda term A into a graph [ A ] as described previously (multiple transformations are possible because of the choice of Υ trees). Then: - (a) for any term A the graph [ A ] is in λGRAPH , - (b) if [ A ] ′ and [ A ]' are transformations of the term A then we may pass from [ A ] ′ to [ A ]' by using a finite number (exponential in the number of leaves of the syntactic tree of A ) of (CO-ASSOC) moves, - (c) if B is obtained from A by α -conversion then we may pass from [ A ] to [ B ] by a finite sequence of (CO-ASSOC) moves, - (d) let A,B ∈ T ( X ) be two terms and x ∈ X be a variable. Consider the terms λx.A and A [ x := B ] , where A [ x := B ] is the term obtained by substituting in A the free occurrences of x by B . We know that β reduction in lambda calculus consists in passing from ( λx.A ) B to A [ x := B ] . Then, by one β move in GRAPH applied to [( λx.A ) B ] we pass to a graph which can be further transformed into one of A [ x := B ] , via (global FAN-OUT) moves, (CO-ASSOC) moves and pruning moves, - (e) with the notations from (d), consider the terms A and λx.Ax with x ∈ FV ( A ) ; then the η reduction, consisting in passing from λx.Ax to A , corresponds to the ext1 move applied to the graphs [ λx.Ax ] and [ A ] . Proof. (a) we have to prove that for any node λ any oriented path in [ A ] starting at the left exiting edge of this node can be completed to a path which either terminates in a graph , or else terminates at the entry peg of this node, but this is clear. Indeed, either the bound variable (of this λ node in the syntactic tree of A ) is fresh, then the bound variable is replaced by a gate, or else, the bound variable is replaced by a tree of Υ gates. No matter which path we choose, we may complete it to a cycle passing by the said λ node. (b) Clear also, because the (CO-ASSOC) move is designed for passing from a tree of Υ gates to another tree with the same number of leaves. (c) Indeed, the names of bound variables of A do not affect the construction of [ A ], therefore if B is obtained by α -conversion of A , then [ B ] differs from [ A ] only by the particular choice of trees of Υ gates. But this is solved by (CO-ASSOC) moves. (d) This is the surprising, maybe, part of the theorem. There are two cases: x is fresh for A or not. If x is fresh for A then in the graph [( λx.A ) B ] the name variable x is replaced by a gate. If not, then all the occurrences of x in A are connected by a Υ tree with root at the left peg of the λ gate where x appears as a bound variable. In the case when x is not fresh for A , we see in the LHS of the figure the graph [( λx.A ) B ] (with a remanent decoration of 'x'). We perform a graphic ( β ) move and we obtain the graph from the right. <details> <summary>Image 28 Details</summary>  ### Visual Description \n ## Diagram: Molecular Orbital Energy Level Diagram ### Overview The image depicts a molecular orbital energy level diagram for a diatomic molecule, showing the interaction of atomic orbitals to form bonding and antibonding molecular orbitals. Two diagrams are presented side-by-side, connected by a double-headed arrow labeled "β", indicating a potential transformation or relationship between the two configurations. The diagrams illustrate the filling of molecular orbitals with electrons. ### Components/Axes The diagram consists of the following components: * **Atomic Orbitals:** Represented by circles labeled 'A' and 'λ' (lambda). 'A' has vertical lines extending downwards, suggesting s-character. 'λ' is a more complex orbital shape. * **Molecular Orbitals:** Represented by energy levels indicated by numbers 1 through 4, and a higher energy level at the top (labeled 3). * **Electrons:** Represented by arrows within the orbitals, indicating spin. * **Diatomic Molecule:** The overall structure represents a diatomic molecule with atoms labeled 'A' and 'B'. * **Arrows:** Indicate the energy levels and orbital interactions. * **Dashed Lines:** Show the mixing of atomic orbitals to form molecular orbitals. * **Label 'x':** Indicates the initial state of the atomic orbitals. * **Label 'β':** Indicates a transformation or relationship between the two diagrams. ### Detailed Analysis or Content Details **Left Diagram:** * **Atom A:** Contains two electrons in the orbital with vertical lines (s-orbital). * **Atom λ:** Contains two electrons in the orbital labeled 'λ'. * **Molecular Orbital 1:** Filled with two electrons (opposite spins). * **Molecular Orbital 2:** Filled with two electrons (opposite spins). * **Molecular Orbital 3:** Empty. * **Molecular Orbital 4:** Contains two electrons (opposite spins). * **Energy Level 3:** Shows a higher energy level with a circular representation, indicating a non-bonding or antibonding orbital. * **Arrow B:** Points to the higher energy level. **Right Diagram:** * **Atom A:** Contains two electrons in the orbital with vertical lines (s-orbital). * **Atom λ:** Contains two electrons in the orbital labeled 'λ'. * **Molecular Orbital 1:** Filled with two electrons (opposite spins). * **Molecular Orbital 2:** Empty. * **Molecular Orbital 3:** Filled with two electrons (opposite spins). * **Molecular Orbital 4:** Contains two electrons (opposite spins). * **Energy Level 3:** Shows a higher energy level with a circular representation, indicating a non-bonding or antibonding orbital. * **Arrow B:** Points to the higher energy level. ### Key Observations * The diagrams show a change in the electron distribution within the molecular orbitals. In the left diagram, orbitals 1, 2, and 4 are filled, while in the right diagram, orbitals 1, 3, and 4 are filled. * The transformation indicated by "β" suggests a shift of electrons from orbital 2 to orbital 3. * The diagrams represent different electronic configurations of the diatomic molecule. * The 'λ' orbital appears to be involved in the formation of the molecular orbitals. ### Interpretation The diagram illustrates the concept of molecular orbital formation and electron configuration in a diatomic molecule. The "β" transformation likely represents an excitation or rearrangement of electrons within the molecular orbitals, potentially due to an external stimulus like light or heat. The change in electron distribution affects the molecule's energy and properties. The diagram suggests that the molecule can exist in different electronic states, each with its own energy level and stability. The filling of the molecular orbitals follows Hund's rule and the Pauli exclusion principle. The diagram is a simplified representation of a complex quantum mechanical phenomenon, but it effectively conveys the basic principles of molecular orbital theory. The difference in the diagrams suggests a change in the molecule's electronic state, potentially related to a transition between bonding and antibonding orbitals. </details> This graph can be transformed into a graph of A [ x := B ] via (global FAN-OUT) and (CO-ASSOC) moves. The case when x is fresh for A is figured next. <details> <summary>Image 29 Details</summary>  ### Visual Description \n ## Diagram: Mechanism with Linkages ### Overview The image depicts a mechanical linkage system with two configurations, separated by a bidirectional arrow labeled "β". The system consists of a fixed base (labeled 'A'), a series of links, and a circular component with an applied force. The diagram illustrates how the system transforms between two states, likely representing different stages of motion or force application. ### Components/Axes The diagram features the following components: * **A:** Fixed base, positioned at the bottom-left of each configuration. * **Links:** Four links are numbered 1, 2, 3, and 4. * **λ (Lambda):** A circular node representing a joint or connection point. * **B:** A point where a force is applied. * **β:** Bidirectional arrow indicating a transformation or change in configuration. * **Vertical Force:** An arrow pointing upwards, indicating an applied vertical force. * **Dotted Lines:** Represent the range of motion or possible positions of the links. There are no explicit axes in this diagram. The positioning of the components implies a spatial relationship. ### Detailed Analysis or Content Details **Left Configuration:** * Link 1 connects point A to the circular node λ. * Link 2 connects λ to point 3. * Link 3 extends vertically upwards, with the applied vertical force acting on it. * Link 4 connects point 3 to point B. * The circular node λ has an arrow indicating a rotational direction. * The dotted lines show the possible range of motion for links 1 and 2. **Right Configuration:** * Link 1 connects point A to a point below λ. * Link 2 connects to a point above λ. * Link 3 extends vertically upwards, with the applied vertical force acting on it. * Link 4 connects point 3 to point B. * The dotted lines show the possible range of motion for links 1 and 2. The arrow "β" is positioned horizontally between the two configurations, indicating a transformation. ### Key Observations * The system appears to convert a vertical force into a horizontal displacement at point B. * The configuration changes significantly with the transformation indicated by β. * The circular node λ acts as a pivot point, changing its position and orientation between the two configurations. * The dotted lines suggest a degree of freedom or range of motion for the links. ### Interpretation This diagram likely represents a four-bar linkage mechanism. The transformation indicated by β suggests a change in the input or output of the mechanism. The vertical force applied at point 3 is likely the driving force, and the resulting motion at point B could be used to perform work. The diagram demonstrates how a linkage system can convert one type of motion (vertical force) into another (horizontal displacement). The change in configuration suggests a cyclical or reciprocating motion. The presence of λ indicates a rotational joint, and the dotted lines show the range of possible movements. The diagram is a simplified representation of a mechanical system, focusing on the key components and their relationships. It does not provide quantitative data, but rather illustrates the principle of operation. </details> We see that the graph obtained by performing the graphic ( β ) move is the union of the graph of A and the graph of B with a gate added at the root. By pruning we are left with the graph of A , which is consistent to the fact that when x is fresh for A then ( λx.A ) B transforms by β reduction into A . (e) In the next figure we see at the LHS the graph [ λx.Ax ] and at the RHS the graph [ A ]. <details> <summary>Image 30 Details</summary>  ### Visual Description \n ## Diagram: Abstract Computational Model ### Overview The image depicts an abstract diagram representing a computational model, likely related to lambda calculus or a similar formal system. It consists of a circular structure with labeled nodes and arrows indicating flow or transformation. There's also a bidirectional arrow labeled "ext 1" connecting the circular structure to a vertical arrow labeled "A". ### Components/Axes The diagram contains the following components: * **Nodes:** * A node labeled "λ" (lambda) at the top. * A node containing a symbol resembling a "Y" inside a circle. * A node labeled "A" within the circular structure. * A small asterisk (*) positioned to the left of the circular structure. * **Arrows:** * Arrows originating from the "λ" node pointing downwards towards the circular structure. * Arrows within the circular structure connecting the "λ" node to the "Y" node and the "Y" node to the "A" node, and the "A" node back to the "λ" node, forming a loop. * A bidirectional arrow labeled "ext 1" connecting the circular structure to a vertical arrow. * A vertical arrow labeled "A" on the right side of the diagram. ### Detailed Analysis / Content Details The diagram shows a closed loop starting at "λ", going to the "Y" node, then to "A", and back to "λ". The asterisk (*) is positioned outside the loop, near the starting point of the loop. The "ext 1" arrow suggests an external operation or transformation applied to the circular structure, resulting in the "A" output. The "Y" node is a complex symbol, resembling a stylized "Y" within a circle. It's unclear what this symbol represents without further context. ### Key Observations * The circular structure represents a recursive or iterative process. * The "ext 1" operation appears to extract or transform the result of the loop into a single value "A". * The asterisk (*) might indicate an initial state or input to the system. * The diagram is highly abstract and lacks numerical data or specific values. ### Interpretation This diagram likely represents a computational process, possibly a fixed-point computation or a recursive function definition. The "λ" node could represent a lambda abstraction, defining a function. The "Y" node might represent a combinator, a higher-order function used to achieve recursion. The loop represents the repeated application of the function until a fixed point is reached. The "ext 1" operation then extracts the final result, which is labeled "A". The asterisk (*) could represent the initial input or seed value for the computation. The diagram suggests a process where a function is applied repeatedly to itself, ultimately converging to a stable output "A". The "ext 1" operation is crucial for obtaining the final result from the iterative process. Without additional context, it's difficult to determine the precise meaning of the diagram. However, it clearly illustrates a computational model involving recursion, abstraction, and transformation. The diagram is a visual representation of a mathematical or logical concept, rather than a presentation of empirical data. </details> The red asterisk marks the arrow which appears in the construction [ λx.Ax ] from the variable x , taking into account the hypothesis x ∈ FV ( A ). We have a pattern where we can apply the ext1 move and we obtain [ A ], as claimed. As an example, let us manipulate the graph of Ω = ( λx. ( xx ))( λx. ( xx )): <details> <summary>Image 31 Details</summary>  ### Visual Description \n ## Diagram: Network Flow with Fan-Out and Beta Relationship ### Overview The image depicts a diagram illustrating a network flow with a concept labeled "global FAN-OUT" and a relationship denoted by "β". The diagram consists of two main network structures connected by bidirectional arrows. Each network structure is composed of nodes connected by directed edges, with some nodes labeled "λ". The diagram appears to be illustrating a process of branching or distribution within a network. ### Components/Axes The diagram contains the following components: * **Nodes:** Represented by circles. Some nodes are labeled with the Greek letter "λ". * **Edges:** Represented by arrows, indicating the direction of flow. * **Bidirectional Arrows:** Labeled "β" and "global FAN-OUT", indicating relationships or transformations between the two network structures. * **Dashed Circles:** Enclose portions of the network structures, potentially highlighting specific sub-networks or areas of interest. * **Branching Nodes:** Nodes with multiple outgoing edges, representing a split or distribution of flow. ### Detailed Analysis or Content Details The diagram shows two distinct network structures. **Left Network Structure:** * The top node has three outgoing edges. * Below this, there are two nodes labeled "λ". * Each "λ" node has three outgoing edges. * The bottom nodes have no outgoing edges, indicating they are terminal points. * A dashed circle encompasses the top node and the two "λ" nodes. **Right Network Structure:** * The bottom node has three outgoing edges. * Above this, there is a node labeled "λ". * The "λ" node has three outgoing edges. * The top node has no outgoing edges, indicating it is a terminal point. * A dashed circle encompasses the "λ" node and the top node. **Relationships:** * An arrow labeled "β" points from the left network structure to the right network structure. * An arrow labeled "global FAN-OUT" points from the right network structure to the left network structure. ### Key Observations * The "λ" nodes appear to be central to the branching process in both networks. * The dashed circles highlight the core components of each network. * The bidirectional arrows suggest a cyclical or iterative relationship between the two network structures. * The "global FAN-OUT" label suggests a process of distributing flow from a single source to multiple destinations. ### Interpretation The diagram likely represents a model of a system where information or resources are distributed through a network. The "λ" nodes could represent decision points or distribution centers. The "β" relationship might represent a feedback loop or a transformation process. The "global FAN-OUT" indicates a mechanism for broadcasting or replicating information across the network. The cyclical nature of the arrows suggests that the system is dynamic and self-regulating. The diagram could be used to model various systems, such as: * **Computer Networks:** Where data is routed from a source to multiple destinations. * **Biological Systems:** Where signals are transmitted through a network of neurons. * **Social Networks:** Where information is spread through a network of individuals. The diagram is abstract and does not provide specific numerical data. It is a conceptual representation of a network flow process. The use of Greek letters and specialized terminology ("FAN-OUT") suggests a technical context, potentially related to computer science, engineering, or biology. </details> We can pass from the LHS figure to the RHS figure by using a graphic ( β ) move. Conversely, we can pass from the RHS figure to the LHS figure by using a (global FAN-OUT) move. These manipulations correspond to the well known fact that Ω is left unchanged after β reduction: let U = λx. ( xx ), then Ω = UU = ( λx. ( xx )) U ↔ UU = Ω. ## 3.1 Example: combinatory logic S , K and I combinators in GRAPH . The combinators I = λx.x , K = λx. ( λy. ( xy )) and S = λx. ( λy. ( λz. (( xz )( yz )))) have the following correspondents in GRAPH , denoted by the same letters: <details> <summary>Image 32 Details</summary>  ### Visual Description \n ## Diagram: Lambda Calculus Representations ### Overview The image presents three diagrams representing different expressions within lambda calculus. Each diagram consists of nodes connected by directed edges, visually depicting function application and abstraction. The diagrams are labeled 'I', 'K', and 'S' at the bottom, likely representing identity, constant, and substitution combinators respectively. ### Components/Axes The diagrams do not have traditional axes or scales. They are composed of: * **Nodes:** Represented by circles, some containing the symbol 'λ' (lambda). * **Directed Edges:** Arrows indicating the flow of function application. * **Labels:** 'I', 'K', and 'S' identifying each diagram. ### Detailed Analysis or Content Details **Diagram I (Identity Combinator):** * A single circular node with a self-looping arrow. * The arrow originates and terminates at the same node. * This represents the identity function, which returns its input unchanged. **Diagram K (Constant Combinator):** * Two circular nodes, both labeled with 'λ'. * An arrow points from the left 'λ' node to the right 'λ' node. * An arrow loops back from the right 'λ' node to the left 'λ' node. * A vertical line originates from the right 'λ' node, indicating an input. * This represents a function that takes two arguments and returns the first argument, ignoring the second. **Diagram S (Substitution Combinator):** * Three 'λ' nodes in a horizontal sequence, with arrows connecting them from left to right. * The leftmost 'λ' node has an arrow pointing downwards, representing an input. * From the rightmost 'λ' node, three arrows diverge: * One arrow points downwards, representing an output. * Two arrows point to two 'Y'-shaped nodes. * Each 'Y' node has two outgoing arrows, both pointing to the output node. * A curved arrow connects the output node back to the leftmost 'λ' node. * This represents a function that applies a function to an argument, substituting the argument into the function. ### Key Observations * The diagrams visually represent the core concepts of lambda calculus: function abstraction (λ) and application (arrows). * The complexity of the diagrams increases from 'I' to 'K' to 'S', reflecting the increasing complexity of the corresponding combinators. * The 'S' combinator is significantly more complex, indicating its role as a fundamental building block for more complex functions. ### Interpretation These diagrams are visual representations of fundamental combinators in lambda calculus. Lambda calculus is a formal system in mathematical logic and computer science for expressing computation based on function abstraction and application. * **I (Identity):** The simplest combinator, it demonstrates the basic principle of returning an input value. * **K (Constant):** This combinator highlights the ability to ignore an input and return a fixed value. * **S (Substitution):** The most powerful of the three, 'S' demonstrates the core mechanism of function application and substitution, allowing for the construction of more complex functions from simpler ones. The diagrams demonstrate how complex computational behavior can be built from a small set of primitive operations. The 'S' combinator, in particular, is crucial because it allows for the construction of any computable function using only itself and the 'K' and 'I' combinators. This illustrates the Church-Turing thesis, which states that any effectively calculable function can be expressed in lambda calculus. The diagrams are not providing numerical data, but rather a symbolic representation of computational logic. They are a visual aid for understanding the abstract concepts of lambda calculus. </details> Proposition 3.2 (a) By one graphic ( β ) move I A transforms into A , for any A ∈ GRAPH with one output. (b) By two graphic ( β ) moves, followed by a global pruning, for any A,B ∈ GRAPH with one output, the graph ( K A ) B transforms into A . (c) By five graphic ( β ) moves, followed by one local pruning move, the graph ( S K ) K transforms into I . (d) By three graphic ( β ) moves followed by a (global FAN-OUT) move, for any A,B,C ∈ GRAPH with one output, the graph (( S A ) B ) C transforms into the graph ( A C ) ( B C ) . Proof. The proof of (b) is given in the next figure. <details> <summary>Image 33 Details</summary>  ### Visual Description \n ## Diagram: Pruning Process Illustration ### Overview The image depicts a series of diagrams illustrating a pruning process, likely within a computational or algorithmic context. The diagrams show a transformation from a complex network structure to a simplified one, with labels indicating the steps and elements involved. The process appears to involve the removal of components based on parameters 'A' and 'B', and a 'global pruning' step. ### Components/Axes The diagrams consist of nodes (represented as circles) connected by directed edges (arrows). The following labels are present: * **A**: Appears at the bottom-right, and also as a label for an edge. * **B**: Appears at the top-right, and also as a label for an edge. * **λ (lambda)**: Appears within circles, representing a node type. * **β (beta)**: Appears between the diagrams, indicating a transformation step. * **global pruning**: Appears vertically on the right side, indicating a pruning operation. * Dashed circle: Highlights a portion of the middle diagram. ### Detailed Analysis or Content Details The diagrams can be broken down into three stages, moving from left to right: **Stage 1 (Leftmost Diagram):** * A complex network with multiple nodes. * Two nodes are labeled 'λ'. * An edge labeled 'A' connects to a node. * An edge labeled 'B' connects to a node. * A loop connects the two 'λ' nodes. * A dashed line connects the two 'λ' nodes. **Stage 2 (Middle Diagram):** * A simplified network compared to Stage 1. * A node labeled 'λ' is present. * An edge labeled 'A' connects to the 'λ' node. * An edge labeled 'B' connects to the 'λ' node. * The 'λ' node is enclosed in a dashed circle. * The transformation from Stage 1 to Stage 2 is indicated by 'β'. **Stage 3 (Rightmost Diagram):** * A further simplified network. * Two edges, labeled 'A' and 'B', point upwards. * An arc connects the end of edge 'B' back to the start of edge 'A'. * The 'global pruning' operation is indicated by a double-headed arrow. ### Key Observations * The diagrams demonstrate a progressive simplification of a network structure. * The 'β' transformation appears to reduce the complexity of the network. * The 'global pruning' step further simplifies the network, potentially removing redundant or less important components. * The 'λ' nodes seem to be central to the pruning process, as they are present in the initial stages and are removed or simplified in subsequent stages. * The dashed circle in Stage 2 highlights a specific portion of the network that is being targeted for pruning. ### Interpretation The diagram illustrates a pruning algorithm or process used to reduce the complexity of a network. The parameters 'A' and 'B' likely represent criteria or thresholds used to determine which components to remove. The 'β' transformation represents a step in the pruning process, and the 'global pruning' step represents a final refinement. The use of 'λ' nodes suggests that these nodes are particularly susceptible to pruning, potentially representing redundant or less important elements in the network. The overall process aims to simplify the network while preserving its essential functionality. The diagram does not provide quantitative data, but rather a visual representation of the pruning process. The diagram suggests a hierarchical pruning approach, where initial pruning steps (β) are followed by a more global pruning operation. The dashed circle indicates a focus on specific network components during the pruning process. </details> The proof of (c) is given in the following figure. (a) and (d) are left to the interested reader. <details> <summary>Image 34 Details</summary>  ### Visual Description \n ## Diagram: Network Pruning Illustration ### Overview The image presents a series of diagrams illustrating a network pruning process. The diagrams depict directed graphs with nodes and edges, showing how the network structure is modified through operations labeled "twice β", "β", and "local pruning". The diagrams demonstrate a reduction in network complexity over several steps. ### Components/Axes The diagrams consist of nodes represented by circles. Some circles contain the Greek letter lambda (λ) and some contain the Greek letter gamma (γ). Arrows indicate the direction of connections between nodes. Labels "K", "twice β", "β", and "local pruning" describe the transformations applied to the network. There are no explicit axes or scales. ### Detailed Analysis or Content Details **Diagram 1 (Top-Left):** * The diagram shows a network with multiple nodes. * Two nodes at the top are labeled "K". * Several nodes are labeled "λ". * One node is labeled "γ". * Connections are primarily downward and to the right, with some feedback loops. **Diagram 2 (Top-Right):** * This diagram is a transformation of Diagram 1, labeled "twice β". * The network structure is more complex than Diagram 1, with additional connections and loops. * Nodes are labeled "λ" and "γ" as in Diagram 1. * The connections are more convoluted, with several loops. **Diagram 3 (Bottom-Left):** * This diagram is a transformation of Diagram 1, labeled "local pruning". * The network is significantly simplified compared to Diagram 1. * It consists of a single node labeled "λ" with a self-loop and a downward connection to another node labeled "λ". **Diagram 4 (Bottom-Center):** * This diagram is a transformation of Diagram 3, labeled "β". * The network has been expanded slightly from Diagram 3. * It consists of two nodes labeled "λ" connected by a downward arrow and a feedback loop. * A node labeled "γ" is connected to the first "λ" node. **Diagram 5 (Bottom-Right):** * This diagram is a transformation of Diagram 4, labeled "twice β". * The network is more complex than Diagram 4. * It includes nodes labeled "λ", "γ", and "K". * The connections are more intricate, with several loops and feedback paths. ### Key Observations * The "local pruning" operation drastically reduces the network complexity. * The "β" and "twice β" operations appear to add complexity back into the network, but in a different configuration. * The presence of "K" nodes suggests a potential initialization or constraint within the network. * The diagrams demonstrate a cyclical process of pruning and expansion. ### Interpretation The diagrams illustrate a network pruning and regrowth process, likely used in machine learning or neural network optimization. The "local pruning" step removes connections, simplifying the network. The "β" and "twice β" steps then reintroduce connections, potentially to refine the network's structure or improve its performance. The "K" nodes might represent key parameters or constraints that are maintained throughout the process. The cyclical nature of the process suggests an iterative optimization algorithm. The use of Greek letters (λ and γ) likely represents specific parameters or functions within the network. The diagrams demonstrate a method for reducing network size while attempting to maintain or improve functionality. The process appears to be a form of regularization, preventing overfitting by reducing the number of parameters. The diagrams are conceptual and do not provide specific numerical data, but they effectively convey the overall process of network pruning and regrowth. </details> ## 4 Using graphic lambda calculus The manipulations of graphs presented in this section can be applied for graphs which represent lambda terms. However, they can also be applied for graphs which do not represent lambda terms. Fixed points. Let's start with a graph A ∈ GRAPH , which has one distinguished input and one distinguished output.I represent this as follows. $$\rightarrow A \rightarrow$$ For any graph B with one output, we denote by A ( B ) the graph obtained by grafting the output of B to the input of A . I want to find B such that A ( B ) ↔ B , where ↔ means any finite sequence of moves in graphic lambda calculus. I call such a graph B a fixed point of A . The solution of this problem is the same as in usual lambda calculus. We start from the following succession of moves: <details> <summary>Image 35 Details</summary>  ### Visual Description \n ## Diagram: Network Architecture Illustration ### Overview The image depicts a diagram illustrating a network architecture with concepts of "GLOBAL" and "FAN-OUT". It shows three distinct network configurations, each represented by a series of nodes connected by directed edges. The diagram uses the symbols "λ" and "A" within the nodes, and "β" to indicate a directional relationship. Green dotted circles highlight specific portions of the network in two of the configurations. ### Components/Axes The diagram consists of: * **Nodes:** Represented by circles containing either "λ" or "A". * **Directed Edges:** Arrows indicating the flow or connection between nodes. * **Labels:** "GLOBAL", "FAN-OUT", "β". * **Highlighting:** Green dotted circles around specific network components. ### Detailed Analysis or Content Details The diagram presents three network configurations: **Configuration 1 (Left):** * A node containing "λ" with a self-loop labeled "A". * This node connects to a series of three nodes. The first two are connected in parallel, and both connect to a final node. * The green dotted circle highlights the two parallel nodes and their connection to the final node. **Configuration 2 (Right):** * A node containing "λ" with a self-loop labeled "A". * This node connects to another node containing "λ" with a self-loop labeled "A". * The second node connects to a final node. * The green dotted circle highlights the first two nodes containing "λ". **Configuration 3 (Bottom):** * A node containing "λ" with a self-loop labeled "A". * This node connects to a node labeled "A". * The node labeled "A" connects to another node labeled "A". * An arrow labeled "β" points from the bottom-most node to the upper-right, indicating a directional relationship. The text "GLOBAL" and "FAN-OUT" are positioned between Configuration 1 and Configuration 2, with a double-headed arrow connecting them. ### Key Observations * The symbol "λ" appears in all configurations, often with a self-loop labeled "A". * The green dotted circles emphasize specific sub-networks within Configurations 1 and 2. * The "GLOBAL" and "FAN-OUT" labels suggest a relationship between the first two configurations, potentially indicating a transformation or expansion of the network. * The "β" label suggests a feedback or control mechanism. ### Interpretation The diagram likely represents a conceptual model of a network or system, possibly in the context of signal processing, control systems, or machine learning. The "λ" symbol could represent a parameter or function, while "A" might represent a constant or input. The "GLOBAL" and "FAN-OUT" labels suggest that the first configuration represents a global state, and the second configuration represents a fan-out or expansion of that state. The green circles highlight the portions of the network that are being expanded or distributed. The "β" label indicates a feedback loop or control signal that influences the system. The diagram as a whole suggests a system where a global state is processed and distributed, with feedback control to maintain stability or achieve a desired outcome. The diagram does not provide quantitative data, but rather a qualitative representation of the system's architecture and relationships. It is a schematic illustration, not a data-driven chart. </details> This is very close to the solution, we only need a small modification: <details> <summary>Image 36 Details</summary>  ### Visual Description \n ## Diagram: Network Topology Illustration ### Overview The image presents a diagram illustrating three different network topologies, likely representing variations in data flow or processing pathways. The diagrams consist of interconnected nodes labeled 'A' and 'λ' (lambda), with arrows indicating the direction of flow. The diagrams are arranged in a triangular pattern, with two diagrams positioned at the top and one at the bottom. Arrows labeled "GLOBAL" and "β" connect the diagrams, indicating relationships or transformations between them. Certain sections within the diagrams are highlighted with dotted green outlines. ### Components/Axes The diagram utilizes the following components: * **Nodes:** Represented by circles, labeled 'A' and 'λ'. * **Arrows:** Indicate the direction of flow or connection between nodes. * **Labels:** "GLOBAL", "FAN-OUT", "β". * **Green Outlines:** Highlight specific sections within the diagrams. There are no axes in this diagram. ### Detailed Analysis or Content Details **Diagram 1 (Top-Left):** * The diagram starts with a node labeled 'A'. * 'A' connects to a series of three nodes, the middle one labeled 'λ'. * The nodes are interconnected with circular arrows, creating a loop. * The bottom node connects to an output arrow. * A section containing the 'λ' node and two adjacent nodes is enclosed in a dotted green outline. **Diagram 2 (Top-Right):** * The diagram begins with a node labeled 'A'. * 'A' connects to a series of three nodes, the middle one labeled 'λ'. * The nodes are interconnected with circular arrows, creating a loop. * The bottom node connects to an output arrow. * A section containing the 'λ' node and two adjacent nodes is enclosed in a dotted green outline. * An arrow labeled "GLOBAL" points from this diagram to Diagram 1. **Diagram 3 (Bottom):** * The diagram starts with a node labeled 'λ'. * 'λ' connects to a series of three nodes, the bottom one labeled 'A'. * The nodes are interconnected with circular arrows, creating a loop. * The bottom node connects to an output arrow. * A section containing the 'λ' node and two adjacent nodes is enclosed in a dotted green outline. * An arrow labeled "β" points from this diagram to Diagram 2. * An arrow labeled "FAN-OUT" points from Diagram 1 to Diagram 3. ### Key Observations * All three diagrams share a similar structure, involving nodes 'A' and 'λ' connected in a looped arrangement. * The green outlines consistently highlight a specific section of the network, potentially indicating a critical processing unit or a region of interest. * The arrows "GLOBAL", "FAN-OUT", and "β" suggest a relationship between the diagrams, possibly representing data transformation or flow control. * The "FAN-OUT" arrow indicates that Diagram 1 feeds into Diagram 3. * The "GLOBAL" arrow indicates that Diagram 2 feeds into Diagram 1. * The "β" arrow indicates that Diagram 3 feeds into Diagram 2. ### Interpretation The diagram likely represents different configurations of a network or processing system. The nodes 'A' and 'λ' could represent different types of processing units or data elements. The loops suggest iterative processing or feedback mechanisms. The arrows "GLOBAL", "FAN-OUT", and "β" indicate how these configurations interact with each other. The "FAN-OUT" arrow suggests a branching or distribution of data from Diagram 1 to Diagram 3. The "GLOBAL" arrow suggests a feedback loop or global control mechanism from Diagram 2 to Diagram 1. The "β" arrow suggests a transformation or adjustment of data from Diagram 3 to Diagram 2. The green outlines may highlight a specific component or sub-network that is common to all three configurations, suggesting its importance in the overall system. The diagram could be illustrating different stages of a process, different levels of abstraction, or different optimization strategies. The overall structure suggests a complex system with interconnected components and feedback loops. </details> Grafting, application or abstraction? If the A , B from the previous paragraph were representing lambda terms, then the natural operation between them is not grafting, but the application. Or, in graphic lambda calculus the application it's represented by an elementary graph, therefore AB (seen as the term in lambda calculus which is obtained as the application of A to B ) is not represented as a grafting of the output of B to the input of A . We can easily transform grafting into the application operation. <details> <summary>Image 37 Details</summary>  ### Visual Description \n ## Diagram: State Transition Equivalence ### Overview The image presents a diagram illustrating the equivalence between a set of state transitions represented by vertical arrows and a corresponding state machine diagram. It demonstrates how a simple sequential process can be represented using a more complex state machine with loops and transitions. ### Components/Axes The diagram consists of three main parts: 1. **Left Side:** Two vertical arrows labeled 'A' and 'B', with an equivalence symbol (three horizontal lines) connecting them to the next part. 2. **Center:** A state transition diagram with a single state 'A' and a transition to state 'B'. A curved arrow loops back from 'B' to 'A'. A bidirectional arrow labeled 'β' is present. 3. **Right Side:** A state machine diagram with two states, represented by circles. The top circle is unlabeled, and the bottom circle is labeled 'λ'. The top circle has an outgoing arrow to 'B' and another to the 'λ' state. The 'λ' state has an outgoing arrow to 'A' and a loop back to itself. ### Detailed Analysis or Content Details * **Left Side:** The vertical arrows 'A' and 'B' represent sequential actions or states. The equivalence symbol indicates that the following diagram is equivalent to this sequence. * **Center:** The state 'A' transitions to 'B'. The loop from 'B' back to 'A' indicates a possible return or repetition of the 'A' state. The bidirectional arrow 'β' suggests a possible relationship or transformation between the states. * **Right Side:** The unlabeled top state transitions to both 'B' and the 'λ' state. The 'λ' state transitions to 'A' and has a self-loop, indicating it can remain in the 'λ' state. The 'λ' state is likely an intermediate state or a holding state. ### Key Observations The diagram shows a transformation from a simple sequential process (A, B) to a state machine representation. The state machine introduces the concept of intermediate states ('λ') and loops, allowing for more complex behavior. The 'β' arrow in the center diagram suggests a possible transformation or relationship between the initial states and the state machine. ### Interpretation This diagram likely illustrates a concept in automata theory or formal languages. It demonstrates how a simple sequence of actions can be modeled using a state machine. The 'β' arrow could represent a transformation or a function that maps the initial sequence to the state machine. The state 'λ' could represent an internal state or a temporary holding state within the machine. The diagram suggests that state machines can represent more complex behaviors than simple sequential processes, allowing for loops, branching, and intermediate states. The equivalence shown implies that the state machine behaves in the same way as the original sequence of actions. This is a common technique used in computer science to model and analyze systems with different states and transitions. </details> Suppose that A and B are graphs representing lambda terms, more precisely suppose that A is representing a term (denoted by A too) and it's input represents a free variable x of the term A . Then the grafting of B to A is the term A [ x := B ] and the graph from the right is representing ( λx.A ) B , therefore both graphs are representing terms from lambda calculus. We can transform grafting into something else: <details> <summary>Image 38 Details</summary>  ### Visual Description \n ## Diagram: State Transition Representation ### Overview The image presents a visual comparison of different representations of state transitions, likely within the context of formal language theory or automata theory. It shows an equivalence between a simple notation and a more complex diagrammatic representation, and then a further transformation into a state diagram with labeled transitions. ### Components/Axes The diagram consists of three main parts, separated by equal signs and arrows. * **Leftmost:** A simple notation with "A" and "B" labels. * **Center:** A looped diagram with "A" and "B" labels and a bidirectional arrow labeled "β". * **Rightmost:** A state diagram with two states, labeled "λ" and "A", and transitions labeled "A" and "B". ### Detailed Analysis or Content Details 1. **Leftmost Representation:** * Two upward arrows are labeled "A" and "B" respectively. * An equals sign (=) connects this representation to the next. This indicates equivalence. 2. **Center Representation:** * An upward arrow labeled "A". * A curved arrow looping back to itself, labeled "A". * An upward arrow labeled "B". * A bidirectional arrow labeled "β" connects this representation to the next. This suggests a transformation or mapping. 3. **Rightmost Representation (State Diagram):** * Two circular states. * The top state is labeled "λ". * The bottom state is labeled "A". * An arrow originates from "λ" and points to "A", labeled "A". * An arrow originates from "λ" and points downwards, labeled "B". * A curved arrow loops back from "A" to "A", labeled "A". * A curved arrow loops back from "A" to "A", with a small arc indicating a transition. ### Key Observations * The diagram illustrates a progression from a concise notation to a more explicit state diagram. * The "β" symbol likely represents a transformation rule or operation. * The state diagram shows two states, "λ" and "A", with transitions based on inputs "A" and "B". * The loop on state "A" indicates that the state can return to itself upon receiving input "A". ### Interpretation This diagram likely demonstrates a method for converting a simple representation of a language or grammar into a formal state machine. The leftmost notation could represent a set of production rules or a simple grammar. The center representation, with the "β" transformation, might be an intermediate step in the conversion process. The rightmost state diagram is a standard representation of a finite automaton, which can be used to recognize strings generated by the original grammar. The "λ" state could represent the initial state, and the transitions define how the automaton moves between states based on the input symbols "A" and "B". The diagram suggests a formal process for representing and analyzing languages or grammars using automata theory. The equivalence indicated by the "=" sign is crucial, implying that the different representations are functionally identical. </details> This has no meaning in lambda calculus, but excepting the orientation of one of the arrows of the graph from the right, it looks like if the abstraction gate (the λ gate) plays the role of an application operation. Zippers and combinators as half-zippers. Let's take n ≥ 1 a natural number and let's consider the following graph in GRAPH , called the n-zipper: <details> <summary>Image 39 Details</summary>  ### Visual Description \n ## Diagram: Quantum Circuit Representation ### Overview The image presents two equivalent representations of a quantum circuit. The left side shows a circuit diagram using standard quantum gate notation, while the right side depicts the same circuit using a different visual style, resembling a block diagram. An equality sign ("≡") connects the two representations, indicating their equivalence. ### Components/Axes The diagram utilizes the following components: * **Input/Output Lines:** Labeled 'B' at the bottom and 'B'' at the top. * **Intermediate Lines:** Labeled 'A₁', 'A₂', ..., 'A_n' at the bottom and 'A'₁', 'A'₂', ..., 'A'_n' at the top. * **Merging/Splitting Points:** Represented by 'Y' shaped junctions. * **Lambda (λ) Symbols:** Representing a specific quantum operation or gate. * **Rectangular Blocks:** In the right-hand side diagram, these represent the same operations as the lambda symbols and merging/splitting points on the left. ### Detailed Analysis or Content Details **Left Side (Standard Quantum Circuit):** The circuit starts with a single input line 'B'. This line splits into 'n' lines labeled 'A₁' through 'A_n' via lambda (λ) symbols. Each 'A_i' line then converges with a corresponding 'A'_i' line at another 'Y' junction. Finally, all 'A'_i' lines merge into a single output line 'B''. **Right Side (Block Diagram):** The right side shows a series of rectangular blocks stacked vertically. The input 'B' enters the bottom block, and the output 'B'' exits the top block. Each block has two input/output lines: 'A_i' and 'A'_i'. The blocks are arranged in a sequence, with 'A₁' and 'A'₁' connected to the bottom block, 'A₂' and 'A'₂' to the middle block, and 'A_n' and 'A'_n' to the top block. The arrangement suggests a parallel processing structure where each 'A_i' line undergoes a transformation represented by the corresponding block, and the results are then combined to produce the output 'B''. ### Key Observations * The two diagrams are visually distinct but represent the same quantum circuit. * The lambda symbols and 'Y' junctions on the left correspond to the rectangular blocks on the right. * The diagram illustrates a process of splitting an input into multiple parallel paths, processing each path independently, and then merging the results. * The subscript notation (A_i, A'_i) indicates that the circuit operates on multiple qubits or quantum channels. ### Interpretation This diagram likely represents a quantum operation that performs a controlled operation on multiple qubits. The splitting and merging of lines suggest a fan-out and fan-in structure, where the input qubit 'B' is applied to multiple qubits 'A_i', and their results are combined to produce the output qubit 'B''. The lambda symbols and blocks likely represent quantum gates or unitary transformations that perform specific operations on the qubits. The equivalence between the two representations highlights the flexibility in visualizing quantum circuits. The standard circuit notation is more common for describing individual gates and their connections, while the block diagram provides a more abstract view of the overall circuit structure. The diagram suggests a parallel processing scheme, which is a common technique in quantum computing to speed up computations. The use of subscripts indicates that the circuit operates on multiple qubits, which is essential for performing complex quantum algorithms. The diagram does not provide specific details about the quantum gates or transformations used, but it illustrates the general structure of a quantum circuit that performs a controlled operation on multiple qubits. </details> At the left is the n-zipper graph; at the right is a notation for it, or a 'macro'. The zipper graph is interesting because it allows to perform (nontrivial) graphic beta moves in a fixed order. In the following picture is figured in red the place where the first graphic beta move is applied. <details> <summary>Image 40 Details</summary>  ### Visual Description \n ## Diagram: Transformation of a Flow Network ### Overview The image depicts a diagram illustrating a transformation of a flow network. Two network configurations are shown, connected by a bidirectional arrow indicating a transformation process. The networks consist of nodes connected by directed edges, with labels indicating the flow variables involved. The diagram appears to represent a mathematical or engineering concept related to network flow or signal processing. ### Components/Axes The diagram consists of two network structures, labeled with variables A1, A2, An, B, A'1, A'2, A'n, and B'. The nodes are represented by circles, and the flow direction is indicated by arrows. The central node in both networks is highlighted with a red circle. A bidirectional arrow connects the two networks, suggesting a transformation or mapping between them. There are no explicit axes or scales. ### Detailed Analysis or Content Details **Left Network:** * The network has a central node where flows A1 and A2 converge, and then split into A'1 and A'2. * Flow B enters from the bottom and splits into A1, A2, and An. * Flows A1, A2, and An converge at the central node. * Flow B' exits from the top. **Right Network:** * Flow B enters from the bottom. * Flows A1 and A2 converge at a central node, splitting into A'1 and A'2. * Flow A'n exits from the top. * Flow A1 is transformed into A'1. * Flow A2 is transformed into A'2. **Transformation:** * The bidirectional arrow indicates a transformation between the two networks. * The transformation appears to involve a change in the flow paths and variable assignments. ### Key Observations * The central node in both networks is highlighted, suggesting its importance in the transformation. * The transformation involves a rearrangement of flow paths and variable assignments. * The diagram does not provide numerical values or specific equations, but rather a conceptual representation of a network transformation. ### Interpretation The diagram likely represents a transformation of a network flow problem. The left network could represent an initial state, and the right network represents a transformed state. The transformation could be a simplification, a change of variables, or a different representation of the same underlying flow problem. The highlighted central node suggests that this node plays a crucial role in the transformation process. The bidirectional arrow indicates that the transformation may be reversible or have an inverse transformation. The diagram is abstract and requires additional context to fully understand the specific meaning of the transformation. It could be related to concepts in linear algebra, graph theory, or network optimization. The diagram is a visual representation of a mathematical or engineering concept, and its purpose is to illustrate the relationship between the two network configurations. The absence of numerical data suggests that the diagram is intended to convey a general principle rather than a specific solution. </details> In terms of zipper notation this graphic beta move has the following appearance: <details> <summary>Image 41 Details</summary>  ### Visual Description \n ## Diagram: Shear Force Distribution in a Rectangular Beam ### Overview The image depicts a schematic diagram illustrating the shear force distribution within a rectangular beam subjected to a vertical load. It shows two views of the beam, one representing the initial state and the other representing a transformed state after considering shear stress. A curved arrow indicates a transformation or relationship between the two views. ### Components/Axes The diagram consists of: * **Rectangular Beam Sections:** Two rectangular sections are shown, representing the beam's cross-section. * **Vertical Forces (B & B'):** Labeled 'B' at the bottom of each beam section, indicating a downward vertical force. 'B'' is also labeled at the top of each section. * **Horizontal Forces (A1, A2, ... An & A'1, A'2, ... A'n):** Labeled 'A1' through 'An' on the left side of the beam and 'A'1 through 'A'n on the right side, representing horizontal shear forces. * **Diagonal Braces:** Diagonal lines within the rectangular sections, representing the shear stress path. * **Transformation Arrow:** A double-headed arrow positioned between the two beam sections, indicating a transformation or relationship. * **Curved Arrow (A1 to A'1):** A curved arrow at the bottom of the diagram, showing the transformation of force A1 to A'1. ### Detailed Analysis / Content Details The diagram illustrates the following: * **Initial State (Left):** The left side shows a rectangular beam section with vertical forces 'B' and 'B'' acting at the top and bottom, respectively. Horizontal shear forces 'A1' through 'An' are applied on the left side, and corresponding forces 'A'1 through 'A'n are applied on the right side. Diagonal braces are present within the rectangle. * **Transformed State (Right):** The right side shows a similar rectangular beam section, but the diagonal braces are more prominent, and the horizontal forces are represented as 'A'1' through 'A'n'. The curved arrow indicates that the horizontal force 'A1' transforms into 'A'1. * **Force Distribution:** The diagram suggests that the vertical shear force 'B' is resolved into horizontal shear forces 'A1' through 'An' and 'A'1 through 'A'n', and these forces are transmitted through the diagonal braces within the beam. * **Transformation:** The transformation indicated by the double-headed arrow and the curved arrow suggests a change in the distribution of shear stresses within the beam. ### Key Observations * The diagram is a simplified representation of shear stress distribution. * The diagonal braces represent the path of shear stress within the beam. * The transformation suggests that the shear stress distribution changes as the load is applied. * The diagram does not provide numerical values for the forces. ### Interpretation The diagram illustrates the concept of shear stress distribution in a rectangular beam. The vertical shear force 'B' is resolved into horizontal shear forces that are transmitted through the diagonal braces. The transformation suggests that the shear stress distribution is not uniform and changes as the load is applied. This diagram is likely used to explain the underlying principles of shear stress in structural mechanics. The diagram is a conceptual illustration and does not provide quantitative data. It is a qualitative representation of how shear forces are distributed within a beam. The curved arrow suggests that the shear force is being redirected or transformed, possibly due to the material properties or geometry of the beam. The diagram is a simplified model and does not account for factors such as stress concentrations or material anisotropy. </details> We see that a n-zipper transforms into a (n-1)-zipper plus an arrow. We may repeat this move, as long as we can. This procedure defines a 'zipper move': <details> <summary>Image 42 Details</summary>  ### Visual Description \n ## Diagram: ZIPn Decomposition ### Overview The image depicts a diagram illustrating the decomposition of a structure labeled "ZIPn" into its constituent parts. The diagram shows a central block with internal connections, and arrows indicating the separation of this block into individual components and their corresponding counterparts. The diagram appears to be a visual representation of a process or algorithm, likely related to data compression or encoding. ### Components/Axes The diagram consists of the following labeled components: * **ZIPn**: A central block representing the overall structure. * **A1, A2, ..., An**: Input components entering the central block. * **A'1, A'2, ..., A'n**: Output components exiting the central block. * **B**: Input component entering the central block from the top. * **B'**: Output component exiting the central block from the top. * Arrows: Indicate the flow and decomposition process. ### Detailed Analysis or Content Details The diagram can be divided into two main sections: the central block and the decomposed components. **Central Block (Left Side):** The central block is a rectangular shape with internal connections represented by lines. It has two sets of input arrows (A1 to An) entering from the left and two sets of output arrows (A'1 to A'n) exiting from the right. Additionally, there is an input arrow B entering from the top and an output arrow B' exiting from the top. The internal structure of the block is not detailed, but it suggests a transformation or processing of the input components. **Decomposed Components (Right Side):** The right side of the diagram shows the decomposed components. Each input-output pair (Ai, A'i) and (B, B') is represented by a short line with arrows at both ends. This suggests a direct mapping or relationship between the input and output components. The ellipsis (...) indicates that there are multiple such pairs between A1 and An. The arrows indicate a flow or transformation from Ai to A'i and from B to B'. The double-headed arrow between the central block and the decomposed components indicates a reversible process or a decomposition/reconstruction relationship. ### Key Observations * The diagram illustrates a decomposition process where a complex structure (ZIPn) is broken down into simpler components. * The input and output components are paired, suggesting a transformation or mapping between them. * The diagram is abstract and does not provide specific numerical values or quantitative data. * The use of primes (e.g., A'1) suggests a modified or transformed version of the original component (A1). ### Interpretation The diagram likely represents a conceptual model of a data compression or encoding algorithm, where "ZIPn" represents the compressed or encoded data. The decomposition process shows how the compressed data can be broken down into its original components (A1 to An and B). The arrows indicate the flow of information during the compression and decompression processes. The diagram suggests that the algorithm involves a transformation of the input components (Ai) into their corresponding output components (A'i), and a similar transformation for the component B into B'. The reversible nature of the process (indicated by the double-headed arrow) implies that the algorithm is lossless, meaning that the original data can be perfectly reconstructed from the compressed data. The diagram is a high-level representation and does not provide details about the specific algorithms or techniques used for compression or encoding. It serves as a visual aid for understanding the overall structure and flow of the process. The diagram is not providing facts or data, but rather a conceptual illustration. </details> We may see the 1-zipper move as the graphic beta move, which transforms the 1-zipper into two arrows. The combinator I = λx.x satisfies the relation IA = A . In the next figure it is shown that I (figured in green), when applied to A , is just a half of the 1-zipper, with an arrow added (figured in blue). <details> <summary>Image 43 Details</summary>  ### Visual Description \n ## Diagram: Quantum Circuit Equivalence ### Overview The image depicts a diagram illustrating the equivalence between two quantum circuits. The diagram shows a circuit on the left side involving a parameter 'λ' and a controlled-NOT (CNOT) gate, and a circuit on the right side involving a different arrangement of gates and feedback loops. The equivalence is indicated by a triple equals sign ("===") between the two circuits. ### Components/Axes The diagram consists of the following components: * **Left Circuit:** * A circle with the label "λ" inside. * A CNOT gate (represented by a circle with a plus sign inside). * Input/Output lines labeled "A". * A green circular feedback loop. * **Right Circuit:** * A square gate with a diagonal line inside (representing a controlled operation). * Input/Output lines labeled "A". * A blue curved feedback loop. * **Equivalence Indicator:** * Three horizontal lines ("===") connecting the two circuits. ### Detailed Analysis or Content Details The left circuit begins with an input "A" entering a CNOT gate. The output of the CNOT gate is then fed back into the input of a gate labeled "λ". The output of the "λ" gate is then fed back into the CNOT gate, forming a closed loop. The right circuit takes an input "A" and feeds it into a square gate. The output of the square gate is then fed back into the input of the same gate, forming a closed loop. The equivalence indicator ("===") suggests that these two circuits perform the same quantum operation. ### Key Observations The diagram highlights a specific equivalence in quantum circuit design. The use of feedback loops in both circuits is notable. The "λ" gate on the left and the square gate on the right represent different quantum operations, but their combination with the feedback loops results in equivalent behavior. ### Interpretation This diagram demonstrates a principle of quantum circuit simplification or transformation. It shows that different arrangements of quantum gates can achieve the same result. This is important for optimizing quantum algorithms and reducing the number of gates required to implement a specific operation. The equivalence likely relies on specific properties of the "λ" gate and the controlled operation within the square gate. The diagram suggests that the parameter "λ" plays a crucial role in establishing the equivalence. The diagram is a visual representation of a mathematical identity in quantum mechanics, showing that two seemingly different quantum operations are, in fact, equivalent. This type of equivalence is often used to simplify quantum circuits and make them more efficient. </details> By opening the zipper we obtain A , as it should. The combinator K = λxy.x satisfies KAB = ( KA ) B = A . In the next figure the combinator K (in green) appears as half of the 2-zipper, with one arrow and one termination gate added (in blue). <details> <summary>Image 44 Details</summary>  ### Visual Description \n ## Diagram: Equivalent Representations of a System ### Overview The image presents two different graphical representations of the same system, indicating their equivalence. The left side shows a block diagram with a rectangular block and input/output lines, while the right side depicts a signal flow diagram with summation junctions and feedback loops. A triple equals sign (=) connects the two representations, signifying their functional equivalence. ### Components/Axes The diagram does not have traditional axes. Instead, it features: * **Block Diagram (Left):** A rectangular block with input labeled "A" and output labeled "B". Lines represent signal flow. A feedback loop is present, originating from the output and returning to the input. * **Signal Flow Diagram (Right):** Summation junctions (circles with a cross inside) and directed arrows representing signal flow. A feedback loop is also present, originating from the output and returning to the input. * **Labels:** "A" and "B" are used to label the input and output signals, respectively. The symbol "λ" (lambda) appears within the signal flow diagram. * **Color Coding:** The block diagram uses blue lines, while the signal flow diagram uses green lines. ### Detailed Analysis or Content Details **Block Diagram (Left):** * Input A enters the block from the right. * Input B enters the block from the top. * Output B exits the block from the top. * A feedback loop originates from the output B, loops around the block, and enters the block from the bottom. **Signal Flow Diagram (Right):** * Input A enters a summation junction. * Input B enters the same summation junction. * The output of the summation junction feeds into another summation junction. * A feedback loop originates from the output of the second summation junction, passes through a component labeled "λ", and returns to the first summation junction. * Another input enters the first summation junction, originating from the left. This input is also passed through a component labeled "λ". ### Key Observations The two diagrams represent the same system functionality but use different graphical conventions. The block diagram is a more abstract representation, while the signal flow diagram provides a more detailed view of the signal paths and summation points. The "λ" symbol likely represents a gain or transfer function within the system. The color coding (blue vs. green) visually separates the two representations. ### Interpretation This diagram illustrates the concept of equivalent representations in system modeling. The block diagram and signal flow diagram are two ways to describe the same system's behavior. The signal flow diagram breaks down the block diagram into its fundamental components: signal addition and feedback. This is a common practice in control systems engineering and signal processing, where different representations are used to analyze and design systems. The presence of the "λ" symbol suggests that the system includes gain elements, which are crucial for understanding the system's response to inputs. The equivalence indicated by the triple equals sign implies that both diagrams accurately predict the system's output for any given input. The diagram is not providing numerical data, but rather a conceptual equivalence. </details> After opening the zipper we obtain a pair made by A and B which gets the termination gate on top of it. A global pruning move sends B to the trash bin. Finally, the combinator S = λxyz. (( xz )( yz )) satisfies SABC = (( SA ) B ) C = ( AC )( BC ). The combinator S (in green) appears to be made by half of the 3-zipper, with some arrows and also with a 'diamond' added (all in blue). Interestingly, the diamond looks alike the ones from the emergent algebra sector, definition 5.4. Expressed with the help of zippers, the relation SKK = I appears like this. <details> <summary>Image 45 Details</summary>  ### Visual Description \n ## Diagram: Network Representation ### Overview The image presents a diagram illustrating two network representations, one on the left in blue and one on the right in green. The two diagrams are visually equated with a triple equals sign ("==="). Both diagrams depict a similar structure involving nodes connected by directed edges, with a central component resembling a processing unit or a series of branching points. The diagrams appear to represent a transformation or equivalence between two different network architectures. ### Components/Axes The diagrams consist of the following components: * **Nodes:** Represented by circles. Some nodes have a "Y" symbol inside, and others have a lambda symbol ("λ"). * **Processing Unit:** A rectangular block with vertical lines inside, representing a series of operations or transformations. * **Directed Edges:** Arrows indicating the flow of information or connections between nodes and the processing unit. * **Labels:** "A", "B", and "C" are used to label the input/output connections to the processing unit. ### Detailed Analysis or Content Details **Left Diagram (Blue):** * The processing unit is positioned centrally. * Three inputs labeled "A", "B", and "C" enter the processing unit from the left. * The output from the processing unit connects to a branching structure consisting of three nodes with "Y" symbols. * A feedback loop connects the output of the branching structure back to the input of the processing unit. * The connections are represented by curved blue lines. **Right Diagram (Green):** * The diagram consists of a series of nodes with lambda symbols ("λ") connected in a chain. * The chain of lambda nodes connects to a branching structure similar to the left diagram, with three nodes containing "Y" symbols. * The inputs labeled "A", "B", and "C" enter the network at the top, connecting to individual lambda nodes. * A feedback loop connects the output of the branching structure back to the chain of lambda nodes. * The connections are represented by curved green lines. ### Key Observations * The diagrams are structurally similar, suggesting an equivalence in functionality. * The processing unit in the left diagram is replaced by a chain of lambda nodes in the right diagram. * The labels "A", "B", and "C" are consistently used in both diagrams, indicating corresponding inputs/outputs. * The feedback loops are present in both diagrams, suggesting a recurrent or iterative process. ### Interpretation The diagram likely represents a transformation of a network architecture. The left diagram, with its processing unit, could represent a more abstract or high-level representation of a computation. The right diagram, with its chain of lambda nodes, could represent a more concrete or low-level implementation of the same computation. The lambda nodes likely represent functional operations, and the chain represents a sequence of these operations. The equivalence indicated by the "===" suggests that the two diagrams are functionally equivalent, meaning they perform the same computation despite having different structures. The feedback loops suggest that the computation is iterative or recurrent, potentially involving learning or adaptation. The labels "A", "B", and "C" likely represent input parameters or signals to the network. The diagram could be illustrating a process of unfolding or expanding a compact representation into a more detailed one, or vice versa. The use of lambda calculus is suggested by the "λ" symbols, indicating a functional programming approach to network design. </details> Lists and currying. With the help of zippers, we may enhance the procedure of turning grafting into the application operation. We have a graph A ∈ GRAPH which has one output and several inputs. <details> <summary>Image 46 Details</summary>  ### Visual Description \n ## Diagram: Pneumatic Circuit Schematic ### Overview The image depicts a series of pneumatic circuit schematics, illustrating a sequential operation involving directional control valves, cylinders, and potentially pressure regulation. The schematics are arranged in a flow-like manner, suggesting a step-by-step process. Each schematic shows a different stage of the circuit's operation, with arrows indicating the flow of air and the sequence of events. ### Components/Axes The diagram utilizes standard pneumatic symbols. Key components include: * **Directional Control Valves:** Represented by square symbols with arrows indicating flow paths. These valves are marked with "K" on the input/output lines. The valves have multiple ports, indicated by the number of arrows. * **Cylinders (Double-Acting):** Represented by rectangles with double-headed arrows indicating piston movement. * **Pilot Valves/Control Lines:** Represented by small circles with "Y" inside, connected to the directional control valves. * **Flow Lines:** Represented by solid lines with arrowheads indicating the direction of airflow. * **Connecting Arrows:** Curved arrows indicating the sequence of operation between the different stages. * **Lambda (λ) Symbol:** Appears within a circular element in one of the schematics. Its function is unclear without additional context. * **Triple Vertical Lines (|||):** Separates two stages of the circuit. ### Detailed Analysis or Content Details The diagram consists of five distinct stages, arranged in a roughly clockwise manner. **Stage 1 (Top-Left):** * A directional control valve with multiple ports (approximately 5 ports) is shown. * Input/output lines are labeled "K". * Two cylinders are connected to the valve, each with a pilot valve ("Y") controlling its operation. * Air flows into the valve from the top and exits to the cylinders. **Stage 2 (Top-Right):** * A directional control valve with multiple ports (approximately 5 ports) is shown. * Input/output lines are labeled "K". * Two cylinders are connected to the valve, each with a pilot valve ("Y") controlling its operation. * A circular element containing the lambda (λ) symbol is connected to the valve. * An arrow connects this stage to Stage 1, indicating a return or feedback loop. **Stage 3 (Center-Left):** * A directional control valve with multiple ports (approximately 5 ports) is shown. * Input/output lines are labeled "K". * Two cylinders are connected to the valve, each with a pilot valve ("Y") controlling its operation. * An arrow connects this stage to Stage 4, indicating a sequential flow. **Stage 4 (Center-Right):** * A directional control valve with multiple ports (approximately 5 ports) is shown. * Input/output lines are labeled "K". * Two cylinders are connected to the valve, each with a pilot valve ("Y") controlling its operation. * An arrow connects this stage to Stage 3, indicating a sequential flow. **Stage 5 (Bottom-Center):** * A directional control valve with fewer ports (approximately 3 ports) is shown. * Input/output lines are labeled "K". * A single cylinder is connected to the valve, with a pilot valve ("Y") controlling its operation. * An arrow connects this stage to Stage 1, completing the sequence. ### Key Observations * The "K" label consistently appears on the input/output lines of the directional control valves, suggesting it represents a common pressure source or control signal. * The pilot valves ("Y") are crucial for controlling the operation of the cylinders. * The sequence of stages suggests a cyclical operation, with the final stage returning to the initial stage. * The lambda (λ) symbol in Stage 2 is an anomaly and its function is unclear. * The number of ports on the directional control valves varies, indicating different levels of control and complexity. ### Interpretation The diagram illustrates a pneumatic circuit designed for a sequential operation, likely involving the coordinated movement of multiple cylinders. The valves control the flow of air to the cylinders, and the pilot valves provide a means of controlling the valves themselves. The cyclical nature of the sequence suggests an automated process, potentially involving repetitive tasks. The "K" label likely represents a compressed air supply or a control signal that initiates the sequence. The lambda symbol could represent a sensor or a specific control function within the circuit, but its exact purpose is unknown without further information. The diagram is a high-level representation of the circuit, and further details would be needed to understand the specific timing and control logic. The diagram is a functional block diagram, and does not provide specific dimensions or performance characteristics. </details> <details> <summary>Image 47 Details</summary>  ### Visual Description \n ## Diagram: Block Diagram with Input/Output ### Overview The image depicts a simple block diagram representing a process or component 'A' with a single input and multiple outputs. The diagram is minimalistic, focusing on the flow of information or material. ### Components/Axes The diagram consists of: * A rectangular block labeled "A" in the center. * A single arrow pointing *into* the top of the block, representing the input. * 'n' arrows pointing *out* from the bottom of the block, representing the outputs, labeled 1, 2, and "..." indicating further outputs up to 'n'. ### Detailed Analysis or Content Details The block labeled 'A' is the central element. The input arrow suggests a single source feeding into this component. The 'n' output arrows indicate that the component 'A' transforms the input into multiple outputs. The outputs are numbered sequentially from 1 to 'n', with an ellipsis indicating that there are more outputs than explicitly shown. The diagram does not provide any quantitative data or specific details about the transformation performed by 'A'. ### Key Observations The diagram is a high-level representation. It does not specify the nature of the input or outputs, nor the function of block 'A'. The use of 'n' suggests a variable or potentially large number of outputs. ### Interpretation This diagram likely represents a branching or distribution process. Component 'A' receives a single input and distributes it into multiple outputs. This could represent a signal splitter, a decision-making process, or a component that generates multiple results from a single input. The diagram is abstract and requires additional context to understand the specific application. The diagram is a simplified model, and does not provide any information about the efficiency, accuracy, or other performance characteristics of the process. It is a conceptual illustration of input-process-output relationship. </details> We use an n-zipper in order to clip the inputs with the output. <details> <summary>Image 48 Details</summary>  ### Visual Description \n ## Diagram: Feedback Loop with Valves and Block A ### Overview The image depicts a schematic diagram of a system featuring a feedback loop. The system includes a series of valves, a block labeled "A", and connecting pathways. The diagram illustrates a closed-loop configuration where output from block A is fed back to influence the input, potentially through the valves. ### Components/Axes The diagram consists of the following components: * **Valves:** Three valves are shown, arranged vertically. They are represented by symbols resembling a valve with an arrow indicating flow direction. They are labeled 1, 2, and 'n' (representing a potentially larger number of valves). * **Block A:** A rectangular block labeled "A" is positioned centrally in the diagram. It represents a processing unit or component within the system. * **Feedback Loop:** Curved lines connect the output of Block A back towards the input, passing through the valves. * **Input:** An arrow at the top of the diagram indicates the system input. * **Ellipsis (...):** Ellipses are used to indicate that the valve series and feedback pathways continue beyond what is explicitly shown. ### Detailed Analysis or Content Details The diagram shows a flow of material or signal entering the system from the top. This flow passes through a series of valves (1, 2, and 'n'). The output from these valves is directed into Block A. Block A then produces an output that is fed back through a looped pathway, again potentially through the valves, to influence the input. The valves appear to be arranged in series, with the input flowing sequentially through each valve before reaching Block A. The feedback loop connects the output of Block A to the input side of the valve series. The ellipsis suggests that there are more valves than just the three explicitly shown. There are no numerical values or specific scales present in the diagram. The diagram is purely schematic, illustrating the relationships between components rather than providing quantitative data. ### Key Observations * The diagram emphasizes a closed-loop control system. * The valves likely regulate the flow or signal strength within the system. * Block A represents a core processing element. * The ellipsis indicates a potentially complex or scalable system. ### Interpretation This diagram likely represents a control system where the output of a process (Block A) is used to adjust the input, maintaining a desired state or condition. The valves act as control elements, modulating the input based on the feedback signal. The use of 'n' suggests that the number of valves can be varied, potentially allowing for fine-tuning of the system's response. The diagram is abstract and does not specify the nature of the process within Block A or the type of signal being controlled. It could represent a variety of systems, such as a temperature control loop, a fluid flow regulation system, or an electronic feedback circuit. The diagram's purpose is to illustrate the fundamental architecture of a closed-loop control system, rather than to provide specific details about its implementation. The diagram is a conceptual representation of a system, and further information would be needed to understand its specific function and behavior. </details> This graph is, in fact, the following one. We may interpret the graph inside the green dotted rectangle as the currying of A , let's call him Curry ( A ). This graph has only one output and no inputs. The graph inside the red dotted rectangle is almost a list. We shall transform it into a list by using again a zipper and one graphic beta move. <details> <summary>Image 49 Details</summary>  ### Visual Description \n ## Diagram: Iterative Process with Feedback Loops ### Overview The image depicts a diagram illustrating an iterative process involving feedback loops and a central processing block labeled "A". The diagram is divided into two main sections, delineated by dashed rectangles: a green section representing a core iterative loop, and a red section representing a sequential progression. The diagram uses circular nodes labeled with the Greek letter lambda (λ) and numerical identifiers (1, 2, n) to represent stages in the process, with arrows indicating the flow of information or control. ### Components/Axes The diagram consists of the following components: * **Circular Nodes:** Represent stages or states in the process. These are labeled with "λ" and numerical identifiers (1, 2, n). * **Block A:** A rectangular block labeled "A", representing a processing unit or function. * **Arrows:** Indicate the direction of flow or feedback within the process. * **Dashed Rectangles:** Define two distinct sections of the diagram: a green iterative loop and a red sequential progression. * **Ellipsis (...):** Indicates continuation of the input arrows to block A. There are no axes or scales present in this diagram. ### Detailed Analysis or Content Details The diagram can be broken down into two sections: **Green Section (Iterative Loop):** * The core of this section consists of a series of circular nodes labeled "λ". These nodes are connected in a loop, with arrows indicating a cyclical flow. * The bottom-most "λ" node feeds into block "A". * Block "A" has multiple input arrows, indicated by an ellipsis (...), suggesting multiple inputs. * Block "A" outputs to the "λ" node above it, creating a feedback loop. * The loop continues with arrows connecting the "λ" nodes in a cascading manner. **Red Section (Sequential Progression):** * This section consists of a series of circular nodes labeled "1", "2", and "n", arranged vertically. * Arrows indicate a sequential flow from node "1" to node "2" to node "n". * The output of the green section (the top-most "λ" node) feeds into node "1" of the red section. **Specific Flow:** 1. The iterative loop (green section) processes information and outputs to node "1" (red section). 2. Node "1" passes the information to node "2". 3. Node "2" passes the information to node "n". 4. Node "n" represents the final stage of the sequential progression. 5. The output of block "A" feeds back into the iterative loop, influencing subsequent iterations. ### Key Observations * The diagram highlights a process that involves both iterative refinement (green section) and sequential progression (red section). * The feedback loop within the green section suggests a process of continuous improvement or adjustment. * The sequential progression in the red section indicates a defined order of operations or stages. * The connection between the two sections suggests that the iterative process informs the sequential progression. ### Interpretation The diagram likely represents a control system or algorithm where a core process (block A) is repeatedly refined through feedback loops (green section) before being applied to a sequential series of steps (red section). The iterative loop allows for adjustments and optimizations based on the output of block A, while the sequential progression ensures a consistent and ordered execution of the final result. The "n" in the red section suggests that the sequential process can be extended to an arbitrary number of steps. The diagram is abstract and does not provide specific details about the nature of the process or the meaning of the labels "λ" and "A". However, it effectively communicates the overall structure and flow of the system. It could represent a machine learning algorithm, a control system in engineering, or a complex data processing pipeline. The diagram emphasizes the importance of feedback and iteration in achieving a desired outcome. </details> Packing arrows. We may pack several arrows into one. I describe first the case of two arrows. We start from the following sequence of three graphic beta moves. <details> <summary>Image 50 Details</summary>  ### Visual Description \n ## Diagram: Functional Programming Illustration ### Overview The image depicts a diagram illustrating a functional programming concept, likely related to currying and list processing. It shows a series of lambda functions chained together, with input from a list and output to a curried function. The diagram is enclosed in a dashed red rectangle, with a green dashed rectangle highlighting a separate component. ### Components/Axes The diagram consists of the following components: * **List(1,2,...,n):** A label indicating an input list containing elements from 1 to n. This is positioned to the left of the dashed red rectangle. * **Lambda Functions (λ):** A series of circular nodes labeled with the Greek letter lambda (λ), representing anonymous functions. These are arranged vertically within the dashed red rectangle. * **Numbers (1, 2, ..., n):** Labels indicating the position of each lambda function in the chain, positioned to the right of each lambda node. * **Curry(A):** A label indicating a curried function, enclosed in a green dashed rectangle on the right side of the diagram. * **Arrows:** Arrows indicating the flow of data between the list, lambda functions, and the curried function. ### Detailed Analysis / Content Details The diagram shows a sequence of lambda functions applied to a list. The list `List(1,2,...,n)` is the initial input. The first element of the list (1) is fed into the first lambda function (λ). The output of this lambda function is then fed into the next lambda function (λ), and so on, until the nth element of the list is processed. The output of the final lambda function is then passed as input to the `Curry(A)` function. The lambda functions are arranged in a vertical chain, with each function taking the output of the previous function as its input. The numbers 1 through n indicate the order in which the elements of the list are processed by the lambda functions. ### Key Observations The diagram illustrates a functional programming technique where a list of values is transformed by applying a series of functions sequentially. The `Curry(A)` function suggests that the final result of the lambda chain is then used as an argument to a curried function. The diagram emphasizes the flow of data and the composition of functions. ### Interpretation This diagram likely demonstrates the application of a function to each element of a list, followed by the application of a curried function to the result. This is a common pattern in functional programming, where functions are treated as first-class citizens and can be passed as arguments to other functions. The currying aspect suggests that the `Curry(A)` function is a function that takes multiple arguments, but is called with only one argument at a time, returning a new function that takes the remaining arguments. The diagram is a visual representation of a functional pipeline, where data flows through a series of transformations. The use of lambda functions suggests that these transformations are simple and concise. The overall purpose of the diagram is to illustrate the power and flexibility of functional programming techniques. The diagram does not contain any numerical data, but rather focuses on the conceptual flow of data and the composition of functions. </details> With words, this figure means: we pack the 1, 2, entries into a list, we pass it trough one <details> <summary>Image 51 Details</summary>  ### Visual Description \n ## Diagram: Network Transformation with Beta Parameter ### Overview The image depicts a series of diagrams illustrating a transformation of a network structure. The diagrams are arranged vertically, showing a progression of changes. Each diagram represents a network with nodes and directed edges. A curved blue arrow labeled "β" appears between the first two and second two diagrams, indicating a transformation process. The diagrams show nodes labeled 1, 2, 3, and 4, and nodes with a symbol resembling a "Y" and the Greek letter lambda (λ). ### Components/Axes The diagrams consist of: * **Nodes:** Represented by circles. Some nodes are labeled with numbers (1, 2, 3, 4) and others with symbols (Y-shaped node and λ). * **Directed Edges:** Arrows indicating the direction of connections between nodes. Edges are shown in black and red. * **Transformation Arrow:** A curved blue arrow labeled "β" indicating a transformation between network states. * **Labels:** Numbers 1, 2, 3, and 4 are used to label input and output nodes. ### Detailed Analysis or Content Details **Diagram 1 (Top):** * Input nodes: 1 and 2. * Node 1 connects to a Y-shaped node. * Node 2 connects to a Y-shaped node. * Both Y-shaped nodes connect to λ nodes. * The λ nodes connect to output nodes 4 and 3 respectively. * There are dotted lines connecting the Y-shaped nodes to the λ nodes. **Diagram 2 (Middle):** * Input nodes: 1 and 2. * Node 1 connects directly to a λ node. * Node 2 connects directly to a λ node. * The λ nodes connect to output nodes 4 and 3 respectively. * Edges are now solid black and red. * The Y-shaped nodes have disappeared. **Diagram 3 (Second from Bottom):** * Input nodes: 1 and 2. * Node 1 connects directly to output node 3. * Node 2 connects directly to a Y-shaped node, which connects to λ node, which connects to output node 4. **Diagram 4 (Bottom):** * Input nodes: 1 and 2. * Node 1 connects directly to output node 3. * Node 2 connects directly to output node 4. ### Key Observations * The network simplifies with each transformation step. * The Y-shaped nodes and dotted lines disappear in the second diagram. * The transformation "β" appears to reduce the complexity of the network. * The final diagram shows a direct connection between input and output nodes. ### Interpretation The diagrams illustrate a process of network simplification or reduction. The parameter "β" likely represents an operation or condition that causes the network to evolve from a more complex state (Diagram 1) to a simpler state (Diagram 4). The initial network (Diagram 1) appears to have intermediate processing stages (Y-shaped and λ nodes) that are eliminated through the transformation "β". The final network (Diagram 4) represents a direct mapping between inputs and outputs, suggesting that the intermediate processing stages are no longer necessary or have been optimized away. The change in edge color from dotted to solid in Diagram 2 could indicate a strengthening or activation of connections. The diagrams could represent a simplification of a computational model, a reduction in the number of layers in a neural network, or a similar process in a complex system. The diagrams are abstract and do not provide specific quantitative data, but they demonstrate a clear trend towards simplification and direct mapping. </details> arrow then we unpack the list into the outputs 3, 4. This packing-unpacking trick may be used of course for more than a pair of arrows, in obvious ways, therefore it is not a restriction of generality to write only about two arrows. We may apply the trick to a pair of graphs A and B , which are connected by a pair of arrows, like in the following figure. <details> <summary>Image 52 Details</summary>  ### Visual Description \n ## Diagram: System Block Representation with Transformation ### Overview The image presents a diagram illustrating a system transformation between two blocks, labeled 'A' and 'B'. The top portion shows a direct, crossed connection between the blocks, while the bottom portion depicts a transformation of this connection using intermediate elements represented by circles with the symbol 'λ' inside. A curved blue arrow between the two sections indicates a transformation or mapping process. ### Components/Axes The diagram consists of: * **Block A:** A rectangular block labeled "A". It has multiple input arrows on the left and output arrows that connect to Block B in the top diagram. * **Block B:** A rectangular block labeled "B". It has multiple input arrows from Block A in the top diagram and multiple output arrows on the right. * **Input Arrows:** Multiple arrows pointing towards Block A, representing inputs. * **Output Arrows:** Multiple arrows originating from Block B, representing outputs. * **Crossed Connections:** Lines connecting the outputs of Block A to the inputs of Block B, crossing each other in the top diagram. * **Lambda (λ) Elements:** Circles containing the symbol 'λ', positioned between Block A and Block B in the bottom diagram. These are connected to both Block A and Block B. * **Transformation Arrow:** A curved blue arrow pointing downwards, indicating a transformation from the top diagram to the bottom diagram. ### Detailed Analysis or Content Details The top diagram shows a direct connection between Block A and Block B. The connections are not explicitly defined in terms of quantity or type, but there are at least two crossing lines. The bottom diagram shows a more complex connection. Block A has multiple outputs that split into three paths, each leading to a 'λ' element. Each 'λ' element then connects to Block B. Block B also has multiple inputs from the 'λ' elements. The number of 'λ' elements appears to be three. The 'λ' symbol suggests a transformation or function being applied to the signal or data. ### Key Observations * The transformation arrow suggests that the bottom diagram represents a decomposition or expansion of the connections shown in the top diagram. * The 'λ' elements likely represent some form of processing or transformation applied to the signals passing between Block A and Block B. * The diagram does not provide any quantitative information about the signals or the transformations. It is a conceptual representation of a system. ### Interpretation The diagram illustrates a system where Block A and Block B are interconnected. The top diagram shows a simplified, direct connection. The bottom diagram demonstrates a more detailed representation of this connection, where the signal flow is mediated by 'λ' elements. These 'λ' elements likely represent functions or transformations applied to the signal. The curved arrow indicates that the bottom diagram is a more refined or expanded view of the relationship between A and B. This could represent a system where Block A generates signals, Block B processes them, and the 'λ' elements represent intermediate processing steps or parameters that influence the signal flow. The use of 'λ' suggests a mathematical or functional transformation. The diagram is abstract and does not specify the nature of the blocks or the transformations, but it provides a conceptual framework for understanding the system's architecture. The diagram is a visual representation of a system's architecture, focusing on the flow of information or signals between components. It doesn't provide specific data or values, but rather a qualitative depiction of the system's structure and relationships. </details> With the added packing and unpacking triples of gates, the graphs A , B are interacting only by the intermediary of one arrow. In particular, we may use this trick for the elementary gates of abstraction and application, transforming them into graphs with one input and one output, like this: <details> <summary>Image 53 Details</summary>  ### Visual Description \n ## Diagram: Network Transformation ### Overview The image presents a visual representation of network transformations. It depicts two initial network structures on the left, a transformation symbol in the center, and the resulting network structures on the right. The diagrams illustrate a process where simpler networks are expanded into more complex ones. ### Components/Axes The diagram consists of: * **Nodes:** Represented by circles, some containing the symbol "λ". * **Arrows:** Indicate the direction of flow or connection between nodes. * **Transformation Symbol:** A green, wavy arrow pointing to the right. * **Initial Networks:** Two networks on the left side of the diagram. * **Resulting Networks:** Two networks on the right side of the diagram. ### Detailed Analysis or Content Details **Top Row:** * **Initial Network:** A single node labeled "λ" with three outgoing arrows. * **Transformation:** The green wavy arrow. * **Resulting Network:** A network consisting of four nodes. One node is labeled "λ". Three other nodes are connected to the "λ" node, and all four nodes have a single outgoing arrow. The connections between the nodes are represented by lines. **Bottom Row:** * **Initial Network:** A single node labeled "λ" with three incoming arrows. * **Transformation:** The green wavy arrow. * **Resulting Network:** A network consisting of four nodes. One node is labeled "λ". Three other nodes are connected to the "λ" node, and all four nodes have a single outgoing arrow. The connections between the nodes are represented by lines. ### Key Observations * The transformation consistently expands a single node network into a network with four nodes. * The symbol "λ" is present in both the initial and resulting networks. * The transformation appears to add complexity to the network structure. * The initial networks differ only in the direction of the arrows (incoming vs. outgoing). ### Interpretation The diagram likely illustrates a process of network expansion or branching. The "λ" symbol could represent a specific element or function within the network. The transformation symbol suggests a process that takes a simple network and creates a more complex one, potentially representing replication, diversification, or the introduction of new elements. The difference between the top and bottom rows suggests that the direction of input/output influences the resulting network structure. The diagram could be used to model processes in various fields, such as biology (gene regulation), computer science (network architecture), or physics (particle interactions). The lack of quantitative data limits a deeper analysis, but the visual representation clearly demonstrates a consistent transformation pattern. </details> If we use the elementary gates transformed into graphs with one input and one output, the graphic beta move becomes this almost algebraic, 1D rule: <details> <summary>Image 54 Details</summary>  ### Visual Description \n ## Diagram: Cascade System with Feedback Loop ### Overview The image depicts a cascade system composed of interconnected nodes, with branching and merging pathways. The system appears to have four input/output points labeled 1 through 4. A feedback loop, denoted by the Greek letter beta (β), is also illustrated. Below the cascade system is a simple cross-shaped diagram representing directional flow. ### Components/Axes The diagram consists of: * **Nodes:** Represented by circles, these are the processing or decision points within the system. * **Arrows:** Indicate the direction of flow or influence between nodes. * **Labels:** Numbers 1, 2, 3, and 4 denote input/output points. The Greek letter lambda (λ) appears multiple times along the pathways. * **Feedback Loop:** A curved arrow labeled β, indicating a cyclical influence. * **Cross Diagram:** A simple diagram with arrows pointing outwards from the center, labeled 1, 2, 3, and 4. ### Detailed Analysis or Content Details The cascade system can be described as follows: 1. **Input 1:** Initiates the flow, splitting into two pathways. 2. **Branching:** The flow splits into two paths, each labeled with λ. 3. **Merging:** The two paths rejoin. 4. **Output 2:** A branch from the system leads to output 2, also labeled with λ. 5. **Cascade Continuation:** The main flow continues, again splitting into two paths labeled with λ. 6. **Further Branching/Merging:** Similar to the first cascade stage, the flow splits and rejoins. 7. **Output 3:** The main flow leads to output 3. 8. **Output 4:** A branch from the system leads to output 4. 9. **Feedback Loop (β):** The curved arrow labeled β connects a point near output 3 back towards the beginning of the cascade, suggesting a feedback mechanism. The cross diagram shows four directional outputs labeled 1, 2, 3, and 4. The arrow pointing towards output 4 is thicker than the others. ### Key Observations * The system is highly interconnected, with multiple branching and merging points. * The lambda (λ) label appears consistently along the pathways, potentially representing a constant or parameter within the system. * The feedback loop (β) suggests a dynamic system where output influences input. * The cross diagram provides a simplified representation of directional flow. ### Interpretation This diagram likely represents a complex system with feedback, possibly a control system or a signal processing network. The cascade structure suggests a series of stages or transformations applied to the input signal. The lambda (λ) label could represent a gain, attenuation, or other transformation parameter at each stage. The feedback loop (β) introduces a dynamic element, allowing the system to adapt or regulate its output. The cross diagram could represent the overall directional influence of the system, with output 4 being the most prominent. The diagram is abstract and doesn't provide specific numerical data, but it illustrates the fundamental structure and relationships within the system. It could be a simplified model of a physical process, a biological pathway, or an engineered system. The diagram is a conceptual representation, and further context would be needed to fully understand its meaning and application. </details> With such procedures, we may transform any graph in GRAPH into a 1D string of graphs, consisting of transformed elementary graphs and packers and un-packers of arrows, which could be used, in principle, for transforming graphic lambda calculus into a text programming language. ## 5 Emergent algebras Emergent algebras [3] [4] are a distillation of differential calculus in metric spaces with dilations [2]. This class of metric spaces contain the 'classical' riemannian manifolds, as well as fractal like spaces as Carnot groups or, more general, sub-riemannian or CarnotCarath´ eodory spaces, Bella¨ ıche [1], Gromov [11], endowed with an intrinsic differential calculus based on some variant of the Pansu derivative [18]. In [2] section 4 Binary decorated trees and dilatations, I propose a formalism for making easy various calculations with dilation structures. This formalism works with moves acting on binary decorated trees, with the leaves decorated with elements of a metric space. Here is an example of the formalism. The moves are (with same names as those used in graphic lambda calculus, see the explanation further): <details> <summary>Image 55 Details</summary>  ### Visual Description \n ## Diagram: Tree Transformation Rules ### Overview The image depicts a diagram illustrating transformation rules for tree structures. It shows four sets of tree diagrams, each representing a rule. Each set consists of a tree on the left, a vertical line in the center, and a transformed tree on the right, connected by a curved blue arrow labeled with a rule identifier (R1a or R2a). The trees are labeled with the characters 'A' and 'B' at their nodes. ### Components/Axes The diagram consists of: * **Trees:** Represented as branching structures with nodes labeled 'A' or 'B'. * **Arrows:** Curved blue arrows indicating the transformation direction. * **Rule Labels:** Text labels above the arrows, identifying the transformation rule (R1a, R2a). * **Node Labels:** Characters 'A' and 'B' labeling the nodes of the trees. * **Vertical Lines:** Representing the input to the transformation. ### Detailed Analysis or Content Details **Rule R1a (Top Row):** * **Left Tree:** A tree with a single node branching into two nodes, both labeled 'A'. * **Vertical Line:** A single vertical line labeled 'A'. * **Right Tree:** A tree with a single node branching into two nodes, both labeled 'A'. A black dot is present at the branching node. **Rule R2a (Bottom Row):** * **Left Tree:** A tree with a single node branching into two nodes. One node is labeled 'A', and the other is labeled 'B'. * **Vertical Line:** A single vertical line labeled 'B'. * **Right Tree:** A tree with a single node branching into two nodes. One node is labeled 'A', and the other is labeled 'B'. A black dot is present at the branching node. ### Key Observations * The rules appear to transform a single node labeled 'A' or 'B' into a tree structure with a branching node and two child nodes. * The branching node in the transformed tree is marked with a black dot. * Rule R1a transforms 'A' into a tree with two 'A' nodes. * Rule R2a transforms 'B' into a tree with 'A' and 'B' nodes. ### Interpretation The diagram illustrates a set of rewrite rules for transforming tree structures. These rules seem to define how to expand a single node labeled 'A' or 'B' into a more complex tree structure. The black dot on the branching node might indicate a specific type of node or a change in its properties during the transformation. The rules suggest a system for building trees from basic elements ('A' and 'B') through a process of expansion or branching. The rules are likely part of a formal grammar or a system for representing and manipulating tree-like data structures. The transformation rules are likely used in a context where the input is a single node and the output is a tree with a branching structure. The rules could be used for parsing, code generation, or other tasks involving tree manipulation. </details> Define the following graph (and think about it as being the graphical representation of an operation u + v with respect to the basepoint x ): <details> <summary>Image 56 Details</summary>  ### Visual Description \n ## Diagram: Tree Representation Equivalence ### Overview The image presents a diagram illustrating the equivalence between two different tree representations. The left side shows a tree with a circled plus sign (+) as the root node, and labeled branches 'u', 'v', and 'x'. The right side shows a tree with a filled circle as the root, and labeled branches 'x', 'u', and 'v'. An equals sign (===) connects the two trees, indicating their equivalence. ### Components/Axes The diagram consists of two tree structures and an equality symbol. The trees are composed of nodes connected by branches. The nodes are labeled with the variables 'u', 'v', and 'x'. The left tree has a circled plus sign (+) at the root, while the right tree has a filled circle at the root. ### Detailed Analysis or Content Details The left tree has a root node labeled '+', enclosed in a circle. Three branches extend from this root: one labeled 'x' extending upwards, one labeled 'u' extending downwards and to the left, and one labeled 'v' extending downwards and to the right. The right tree has a root node represented by a filled black circle. Three branches extend from this root: one labeled 'x' extending upwards, one branching down to a white circle labeled 'u', and another branching down to a white circle labeled 'v'. The equality symbol (===) is positioned horizontally between the two trees, indicating that they represent the same mathematical or logical structure. ### Key Observations The two trees represent the same relationships between the variables 'u', 'v', and 'x', despite having different visual representations for the root node. The left tree uses a circled plus sign, while the right tree uses a filled circle. The order of the branches 'u' and 'v' is reversed in the two trees, but the overall structure remains equivalent. ### Interpretation The diagram demonstrates a transformation or equivalence between two different ways of representing a tree structure. This is likely related to algebraic manipulation or logical simplification. The circled plus sign (+) on the left could represent an addition operation, and the tree structure could represent an expression like x + u + v. The right tree represents the same expression but in a different visual form. The equivalence suggests that the two representations are mathematically or logically identical. The diagram highlights the concept that the same underlying structure can be represented in multiple ways, and that transformations can be performed to change the representation without altering the meaning. The filled and unfilled circles may represent different levels of operation or priority within the tree. </details> Then, in the binary trees formalism I can prove, by using the moves R1a, R2a, the following 'approximate' associativity relation (it is approximate because there appear a basepoint which is different from x , but which, in the geometric context of spaces with dilations, is close to x ): × <details> <summary>Image 57 Details</summary>  ### Visual Description \n ## Diagram: Binary Tree Transformation ### Overview The image depicts a diagram illustrating a transformation between two binary tree structures. Each node in the tree is represented by a circle containing a "+" symbol, indicating a potential operation or value associated with that node. The trees are connected by a bidirectional arrow, suggesting a reversible transformation. The labels 'x', 'y', 'z', and 'u' denote the values or variables associated with the nodes. ### Components/Axes The diagram consists of two binary trees, a bidirectional arrow connecting them, and labels for each node. There are no axes or scales present. The labels are: * x (appears multiple times) * y * z * u * An arc labeled 'u' and 'x' connecting two nodes. ### Detailed Analysis or Content Details The left tree has 'x' as its root. Its left child is 'u', and its right child is a node with a '+' symbol, which has 'y' as its left child and 'z' as its right child. An arc connects the node with 'y' and 'z' as children to the root node 'x', and is labeled 'u' and 'x'. The right tree has 'x' as its root. Its left child is 'z', and its right child is a node with a '+' symbol, which has 'u' as its left child and 'y' as its right child. ### Key Observations The transformation appears to involve swapping the left and right children of the inner nodes. The arc connecting the nodes labeled 'u' and 'x' suggests a relationship or dependency between these values during the transformation. The '+' symbol within each node suggests an operation or calculation performed at that node. ### Interpretation The diagram likely represents a mathematical or computational operation on binary trees. The transformation shown could be a rearrangement or a specific algorithm applied to the tree structure. The '+' symbols suggest that each node performs some operation on its children. The bidirectional arrow indicates that the transformation is reversible, implying an inverse operation exists. The arc labeled 'u' and 'x' could represent a constraint or a dependency that must be maintained during the transformation. Without further context, it's difficult to determine the exact meaning of the transformation, but it likely relates to manipulating tree structures in a specific way. The diagram is abstract and does not provide numerical data, but rather illustrates a structural relationship. </details> It was puzzling that in fact the formalism worked without needing to know which metric space is used. Moreover, reasoning with moves acting on binary trees gave proofs of generalizations of results from sub-riemannian geometry, while classical proofs involve elaborate calculations with pseudo-differential operators. At a close inspection it looked like some- where in the background there is an abstract nonsense machine which is just applied to the particular case of sub-riemannian spaces. In this paper I shall take the following pure algebraic definition of an emergent algebra (compare with definition 5.1 [3]), which is a stronger version of the definition 4.2 [4] of a Γ idempotent right quasigroup, in the sense that here I define a Γ idempotent quasigroup. Definition 5.1 Let Γ be a commutative group with neutral element denoted by 1 and operation denoted multiplicatively. A Γ idempotent quasigroup is a set X endowed with a family of operations ◦ ε : X × X → X , indexed by ε ∈ Γ , such that: - -For any ε ∈ Γ \ { 1 } the pair ( X, ◦ ε ) is an idempotent quasigroup, i.e. for any a, b ∈ X the equations x ◦ ε a = b and a ◦ ε x = b have unique solutions and moreover x ◦ ε x = x for any x ∈ X , - -The operation ◦ 1 is trivial: for any x, y ∈ X we have x ◦ 1 y = y , - -For any x, y ∈ X and any ε, µ ∈ Γ we have: x ◦ ε ( x ◦ µ y ) = x ◦ εµ y . Here are some examples of Γ idempotent quasigroups. Example 1. Real (or complex) vector spaces: let X be a real (complex) vector space, Γ = (0 , + ∞ ) (or Γ = C ∗ ), with multiplication as operation. We define, for any ε ∈ Γ the following quasigroup operation: x ◦ ε y = (1 -ε ) x + εy . These operations give to X the structure of a Γ idempotent quasigroup. Notice that x ◦ ε y is the dilation based at x , of coefficient ε , applied to y . Example 2. Contractible groups: let G be a group endowed with a group morphism φ : G → G . Let Γ = Z with the operation of addition of integers (thus we may adapt definition 5.1 to this example by using ' ε + µ ' instead of ' εµ ' and '0' instead of '1' as the name of the neutral element of Γ). For any ε ∈ Z let x ◦ ε y = xφ ε ( x -1 y ). This a Z idempotent quasigroup. The most interesting case is the one when φ is an uniformly contractive automorphism of a topological group G . The structure of these groups is an active exploration area, see for example [12] and the bibliography therein. A fundamental result here is Siebert [20], which gives a characterization of topological connected contractive locally compact groups as being nilpotent Lie groups endowed with a one parameter family of dilations, i.e. almost Carnot groups. Example 3. A group with an invertible self-mapping φ : G → G such that φ ( e ) = e , where e is the identity of the group G . It looks like Example 2 but it shows that there is no need for φ to be a group morphism. Local versions. We may accept that there is a way (definitely needing care to well formulate, but intuitively clear) to define a local version of the notion of a Γ idempotent quasigroup. With such a definition, for example, a convex subset of a real vector space gives a local (0 , + ∞ ) idempotent quasigroup (as in Example 1) and a neighbourhood of the identity of a topological group G , with an identity preserving, locally defined invertible self map (as in Example 3) gives a Z local idempotent quasigroup. Example 4. A particular case of Example 3 is a Lie group G with the operations defined for any ε ∈ (0 , + ∞ ) by x ◦ ε y = x exp( ε log( x -1 y )). Example 5. A less symmetric example is the one of X being a riemannian manifold, with associated operations defined for any ε ∈ (0 , + ∞ ) by x ◦ ε y = exp x ( ε log x ( y )), where exp is the metric exponential. Example 6. More generally, any metric space with dilations is a local idempotent (right) quasigroup. Example 7. One parameter deformations of quandles. A quandle is a self-distributive quasigroup. Take now a one-parameter family of quandles (indexed by ε ∈ Γ) which satisfies moreover points 2. and 3. from definition 5.1. What is interesting about this example is that quandles appear as decorations of knot diagrams [10] [13], which are preserved by the Reidemeister moves (more on this in the section 6). At closer examination, examples 1, 2 are particular cases of one parameter quandle deformations! I define now the operations of approximate sum and approximate difference associated to a Γ idempotent quasigroup. Definition 5.2 For any ε ∈ Γ we give the following names to several combinations of operations of emergent algebras: - -the approximate sum operation is Σ x ε ( u, v ) = x · ε (( x ◦ ε u ) ◦ ε v ) , - -the approximate difference operation is ∆ x ε ( u, v ) = ( x ◦ ε u ) · ε ( x ◦ ε v ) , - -the approximate inverse operation is inv x ε u = ( x ◦ ε u ) · ε x . Let's see what the approximate sum operation is, for example 1. $$2 ^ { 3 } ( 1 0 , 0 ) = 4 0 ( - 0 ) - 8 + 0$$ It is clear that, as ε converges to 0, this becomes the operation of addition in the vector space with x as neutral element, so it might be said that is the operation of addition of vectors in the tangent space at x , where x is seen as an element of the affine space constructed over the vector space from example 1. This is a general phenomenon, which becomes really interesting in non-commutative situations, i.e. when applied to examples from the end of the provided list. These approximate operations have many algebraic properties which can be found by the abstract nonsense of manipulating binary trees. Another construction which can be done in emergent algebras is the one of taking finite differences (at a high level of generality, not bonded to vector spaces). Definition 5.3 Let A : X → X be a function (from X to itself, for simplicity). The finite difference function associated to A , with respect to the emergent algebra over X , at a point x ∈ X is the following. $$u ) = A ( x ) \cdot e ^ { ( A ( x o _ { e } u ) ) }$$ For example 1, the finite difference has the expression: $$T _ { e } A ( u - x ) = A ( x ) +$$ which is a finite difference indeed. In more generality, for example 2 this definition leads to the Pansu derivative [18]. Finite differences as defined here behave like discrete versions of derivatives. Again, the proofs consist in manipulating well chosen binary trees. All this can be formalized in graphic lambda calculus, thus transforming the proofs into computations inside graphic lambda calculus. I shall not insist more on this, with the exception of describing the emergent algebra sector of graphic lambda calculus. Definition 5.4 For any ε ∈ Γ , the following graphs in GRAPH are introduced: - -the approximate sum graph Σ ε <details> <summary>Image 58 Details</summary>  ### Visual Description \n ## Diagram: State Transition Diagram ### Overview The image depicts a state transition diagram, illustrating the relationships between different states (represented by circles) and the transitions between them (represented by arrows). The diagram appears to model a system with four states: γ, ε, ε⁻¹, and ε. Arrows indicate the direction of state transitions. ### Components/Axes The diagram consists of four circular nodes labeled as follows: * γ (Gamma) - Located on the left side of the diagram. * ε (Epsilon) - Located on the right side of the diagram. * ε⁻¹ (Epsilon inverse) - Located at the top of the diagram. * ε (Epsilon) - Located at the bottom of the diagram. Arrows with arrowheads indicate the direction of transitions between these states. There are no explicit axes or scales. ### Detailed Analysis or Content Details The diagram shows the following transitions: 1. An arrow enters the γ state from the left. 2. An arrow exits the γ state and enters the ε⁻¹ state. 3. An arrow exits the ε⁻¹ state and enters the ε state. 4. An arrow exits the ε state and enters the γ state. 5. An arrow enters the ε state from the right. 6. An arrow exits the ε state and enters the ε state (a self-loop). 7. An arrow exits the ε state and enters the ε⁻¹ state. 8. An arrow exits the ε⁻¹ state and enters the ε state. The diagram does not contain numerical data or quantitative values. It is a qualitative representation of state transitions. ### Key Observations The diagram shows a cyclical flow between the states γ, ε⁻¹, ε, and back to γ. The ε state has a self-loop, indicating that it can remain in that state. The diagram suggests a system where transitions are possible between these states, and the system can cycle through them. ### Interpretation This diagram likely represents a finite state machine or a similar model of a system with discrete states and transitions. The symbols γ and ε suggest mathematical or physical concepts, potentially related to a transformation or operation. The presence of ε⁻¹ indicates an inverse operation. The cyclical nature of the diagram suggests a repeating process or a system that can return to its initial state. The self-loop on ε could represent a stable state or a condition where no change occurs. Without further context, it's difficult to determine the specific meaning of the states and transitions, but the diagram provides a visual representation of their relationships. The diagram is a conceptual model, and its interpretation depends on the specific domain it is applied to. </details> -the approximate difference graph ∆ ε <details> <summary>Image 59 Details</summary>  ### Visual Description \n ## Diagram: State Transition Diagram ### Overview The image depicts a state transition diagram, representing a system with four states: ε (epsilon), ε⁻¹ (epsilon inverse), γ (gamma), and an unspecified initial/final state. Arrows indicate transitions between these states. The diagram is a rhombus shape, with the states positioned at each corner. ### Components/Axes The diagram consists of four states, labeled as follows: * Top: ε⁻¹ * Left: ε * Right: ε * Bottom: γ Arrows indicate transitions between states. There are no explicit axes or scales. The diagram shows a cyclical flow between the states. ### Detailed Analysis or Content Details The diagram shows the following transitions: * An arrow points *into* the ε⁻¹ state from above. * An arrow points *from* the ε⁻¹ state to the ε state (left). * An arrow points *from* the ε state (left) to the γ state (bottom). * An arrow points *from* the γ state (bottom) to the ε state (right). * An arrow points *into* the ε state (right) from the right. * An arrow points *from* the ε state (right) to the ε⁻¹ state (top). The diagram is a closed loop, indicating a continuous cycle of state transitions. The states are connected in a clockwise manner: ε⁻¹ → ε (left) → γ → ε (right) → ε⁻¹. ### Key Observations The diagram represents a system with a cyclical state transition. The presence of ε and ε⁻¹ suggests an inverse operation or a reversible process. The γ state represents a distinct state within the cycle. The diagram does not provide any information about the conditions that trigger the transitions. ### Interpretation This diagram likely represents a mathematical or computational process involving reversible operations. The states could represent different stages in an algorithm, or different values in a variable. The cyclical nature of the diagram suggests that the process repeats indefinitely, or until a specific condition is met. The ε and ε⁻¹ states suggest an operation and its inverse, potentially representing addition and subtraction, or multiplication and division. The γ state could represent a transformation or a different type of operation. Without further context, it is difficult to determine the exact meaning of the diagram. It could be a simplified representation of a more complex system, or a conceptual model of a process. The diagram is a visual representation of a state machine, commonly used in computer science and engineering to model the behavior of systems. </details> -the approximate inverse graph inv ε <details> <summary>Image 60 Details</summary>  ### Visual Description \n ## Diagram: State Transition Diagram ### Overview The image depicts a state transition diagram with three states labeled ε (epsilon), ε⁻¹ (epsilon inverse), and γ (gamma). Arrows indicate transitions between these states, with input and output flows. ### Components/Axes The diagram consists of: * **States:** ε, ε⁻¹, γ * **Transitions:** Arrows connecting the states, indicating flow direction. * **Input/Output:** Arrows entering and exiting the diagram, representing input and output signals. ### Detailed Analysis or Content Details The diagram shows a cyclical flow between the three states. 1. **State ε:** Located on the left side of the diagram. An arrow enters from the top, and an arrow exits to the bottom. An arrow also points from γ to ε. 2. **State ε⁻¹:** Located at the top of the diagram. An arrow enters from ε, and an arrow exits to the top. An arrow also points from γ to ε⁻¹. 3. **State γ:** Located at the bottom of the diagram. An arrow enters from ε⁻¹, and an arrow exits to the bottom. An arrow also points from ε to γ. The arrows indicate a flow where: * Input enters ε, transitions to γ, then to ε⁻¹, and finally back to ε. * Input enters ε⁻¹, transitions to ε, then to γ, and finally back to ε⁻¹. * Input enters γ, transitions to ε, then to ε⁻¹, and finally back to γ. ### Key Observations The diagram represents a closed-loop system with three distinct states and transitions between them. The cyclical nature suggests a repeating process or a system that continuously cycles through these states. ### Interpretation This diagram likely represents a mathematical or computational process involving transformations or operations represented by the states ε, ε⁻¹, and γ. The cyclical nature suggests an iterative process or a system that maintains a state of equilibrium through continuous transitions. The symbols ε and ε⁻¹ suggest an inverse operation, potentially related to an identity or reversal process. The γ state could represent a different type of transformation or a condition that triggers transitions between ε and ε⁻¹. Without further context, it's difficult to determine the specific meaning of these states and transitions, but the diagram clearly illustrates a dynamic system with a defined set of states and transitions. It could be a simplified representation of a more complex system, such as a finite state machine or a control system. </details> Let A be a set of symbols a, b, c, ... . (These symbols will play the role of scale parameters going to 0.) With A and with the abelian group Γ we construct a larger abelian group, call it ¯ Γ, which is generated by A and by Γ. Now we introduce the emergent algebra sector (over the set A ). Definition 5.5 EMER ( A ) is the subset of GRAPH (over the group ¯ Γ ) which is generated by the following list of gates: - -arrows and loops, - -Υ gate and the gates ¯ ε for any ε ∈ Γ , - -the approximate sum gate Σ a and the approximate difference gate ∆ a , for any a ∈ A , with the operations of linking output to input arrows and with the following list of moves: - FAN-OUT moves - -emergent algebra moves for the group ¯ Γ , - -¡pruning moves. The set EMER ( A ) with the given list of moves is called the emergent algebra sector over the set A . The approximate inverse is not included into the list of generating gates. That is because we can prove easily that for any a ∈ A we have inv a ∈ EMER ( A ). (If ε ∈ Γ then we trivially have inv ε ∈ EMER ( A ) because it is constructed from emergent algebra gates decorated by elements in Γ, which are on the list of generating gates.) Here is the proof: we start with the approximate difference ∆ a and with an Υ gate and we arrive to the approximate inverse inv a by a sequence of moves, as follows. <details> <summary>Image 61 Details</summary>  ### Visual Description \n ## Diagram: Co-Association and Rla Transformation ### Overview The image depicts a diagram illustrating a co-association process and a transformation denoted as "Rla". It consists of three interconnected diagrams, with arrows indicating the relationships between them. The diagrams themselves are composed of nodes labeled with "a", "a⁻¹", and "Y", connected by directed edges. ### Components/Axes The diagram contains the following components: * **Nodes:** Labeled "a", "a⁻¹", and "Y". * **Edges:** Directed arrows connecting the nodes, forming cycles. * **Labels:** "CO-ASSOC" and "Rla" are used to label the transformations between the diagrams. * **Diagram 1 (Top-Left):** A triangular structure with nodes "a", "a⁻¹", and "Y" arranged in a cycle. * **Diagram 2 (Top-Right):** A circular structure with nodes "a", "a⁻¹", and "Y" arranged in a cycle. * **Diagram 3 (Bottom-Center):** A simplified structure with nodes "a", "a⁻¹", and "Y" arranged in a cycle. ### Detailed Analysis or Content Details **Diagram 1 (Top-Left):** * The diagram forms a triangle. * Starting from the top, an arrow points from "a⁻¹" to "a". * An arrow points from "a" to "Y". * An arrow points from "Y" to "a". * An arrow points from "a" to "a⁻¹", completing the cycle. **Diagram 2 (Top-Right):** * The diagram forms a circle. * Starting from the top, an arrow points from "a⁻¹" to "a". * An arrow points from "a" to "Y". * An arrow points from "Y" to "a". * An arrow points from "a" to "a⁻¹", completing the cycle. **Diagram 3 (Bottom-Center):** * The diagram forms a triangle. * Starting from the top, an arrow points from "a⁻¹" to "a". * An arrow points from "a" to "Y". * An arrow points from "Y" to "a". **Transformations:** * A bidirectional arrow labeled "CO-ASSOC" connects Diagram 1 and Diagram 2. * A curved arrow labeled "Rla" connects Diagram 3 to Diagram 2. ### Key Observations * Diagrams 1 and 2 share a similar cyclic structure, differing only in the arrangement of the nodes. * Diagram 3 is a simplified version of Diagrams 1 and 2. * The "CO-ASSOC" transformation appears to relate two equivalent representations. * The "Rla" transformation appears to map a simplified structure to a more complex one. ### Interpretation The diagram likely represents a mathematical or computational process involving associative operations and a transformation "Rla". The nodes "a", "a⁻¹", and "Y" could represent elements or variables within this process. The "CO-ASSOC" transformation suggests an equivalence between the two diagrams, possibly representing different ways of expressing the same relationship. The "Rla" transformation indicates a mapping from a simplified representation (Diagram 3) to a more complete one (Diagram 2). The cyclical nature of the diagrams suggests iterative or recursive processes. Without further context, the specific meaning of "a", "a⁻¹", "Y", and the transformations remains unclear, but the diagram provides a visual representation of their relationships. The diagram is a visual representation of an algebraic or logical operation, and the transformations represent steps in a calculation or proof. </details> We proved the following relation for emergent algebras: ∆ x a ( u, x ) = inv x a u . This relation appears as a computation in graphic lambda calculus. As for the finite differences, we may proceed as this. Definition 5.6 A graph A ∈ GRAPH , with one input and one output distinguished, is computable with respect to the group ¯ Gamma if the following graph <details> <summary>Image 62 Details</summary>  ### Visual Description \n ## Diagram: System Flow with Operators ### Overview The image depicts a diagram representing a system flow with two main loops. Each loop contains a circular node labeled with a mathematical operator and a symbol within the circle. Arrows indicate the direction of flow within the loops. Labels "A" and "Y" are placed alongside the loops, indicating input or output. ### Components/Axes The diagram consists of the following components: * **Circular Nodes:** Two circular nodes, one labeled "a" and the other labeled "a⁻¹" (a inverse). * **Symbols within Circles:** Each circular node contains a symbol resembling a "Y" shape. * **Arrows:** Arrows indicating the direction of flow between the nodes and external inputs/outputs. * **Labels:** "A" and "Y" labels positioned alongside the loops. * **Input/Output Lines:** Vertical lines representing input or output to the system. ### Detailed Analysis or Content Details The diagram shows two interconnected loops. * **Loop 1 (Bottom):** * Starts with an input line. * Flows into a circular node containing the symbol "Y" and labeled "a". * Flows out of the "a" node and continues to another input line. * The label "A" is positioned alongside the flow between the input/output lines and the "a" node. * **Loop 2 (Top):** * Starts with an input line. * Flows into a circular node containing the symbol "Y" and labeled "a⁻¹". * Flows out of the "a⁻¹" node and continues to another input line. * The label "A" is positioned alongside the flow between the input/output lines and the "a⁻¹" node. * **Interconnection:** * A horizontal arrow connects the output of the "Y" node in the bottom loop to the input of the "a⁻¹" node in the top loop. * A horizontal arrow connects the output of the "Y" node in the top loop to the input of the "a" node in the bottom loop. ### Key Observations The diagram represents a system where an input is processed by an operator "a" in one loop and its inverse "a⁻¹" in the other. The "Y" symbol within the circles suggests a branching or splitting operation. The loops are interconnected, indicating a feedback or iterative process. ### Interpretation This diagram likely represents a mathematical or signal processing system. The "a" and "a⁻¹" operators suggest a forward and inverse transformation. The "Y" symbol could represent a bifurcation or splitting of a signal. The interconnected loops suggest a system where the output of one transformation is fed back into the other, potentially for iterative refinement or error correction. The "A" label could represent an input signal or a control parameter, while "Y" could represent an output or a branching point. Without further context, it's difficult to determine the specific function of this system, but it appears to be a closed-loop system involving transformations and feedback. The diagram is abstract and does not provide specific numerical data or values. It is a conceptual representation of a system's structure and flow. </details> can be transformed by the moves from graphic lambda calculus into a graph which is made by assembling: - -graphs from EMER ( A ) , - -gates λ , and . It would be interesting to mix the emergent algebra sector with the lambda calculus sector (in a sense this is already suggested in definition 5.6). At first view, it seems that the emergent algebra gates ¯ ε are operations which are added to the lambda calculus operations, the latter being more basic than the former. I think this is not the case. In [5] theorem 3.4, in the formalism of lambda-scale calculus (graphic lambda calculus is a visual variant of this), I show on the contrary that the emergent algebra gates could be applied to lambda terms and the result is a collection, or hierarchy of lambda calculi, organized into an emergent algebra structure. This is surprising, at least for the author, because the initial goal of introducing lambda-scale calculus was to mimic lambda calculus with emergent algebra operations. ## 6 Crossings In this section we discuss about tangle diagrams and graphic lambda calculus. An oriented tangle is a collection of wired in 3D space, more precisely it is an embedding of a oriented one dimensional manifold in 3D space. Two tangles are the same up to topological deformation of the 3D space. An oriented tangle diagram is, visually, a projection of a tangle, in general position, on a plane. More specifically, an oriented tangle diagram is a globally planar oriented graph with 4-valent nodes which represent crossings of wires (as seen in the projection), along with supplementary information about which wire passes over the respective crossing. A locally planar tangle diagram is an oriented graph which satisfies the previous description, with the exception that it is only locally planar. Visually, a locally planar tangle diagram looks like an ordinary one, excepting that there may be crossings of edges of the graph which are not tangle crossings (i.e. nodes of the graph). The purpose of this section is to show that we can 'simulate' tangle diagrams with graphic lambda calculus. This can be expressed more precisely in two ways. The first way is that we can define 'crossing macros' in graphic lambda calculus, which are certain graphs which play the role of crossings in a tangle diagram (i.e. we can express the Reidemeister moves, described further, as compositions of moves from graphic lambda calculus between such graphs). The second way is to say that to any tangle diagram we can associate a graph in GRAPH such that to any Reidemeister move is associated a certain composition of moves from graphic lambda calculus. Meredith ad Snyder [17] achieve this goal with the pi-calculus instead of graphic lambda calculus. Kauffman, in the second part of [14], associates tangle diagrams to combinators and writes about 'knotlogic'. Oriented Reidemeister moves. Two tangles are the same, up to topological equivalence, if and only if any tangle diagram of one tangle can be transformed by a finite sequence of Reidemeister moves into a tangle diagram of the second tangle. The oriented Reidemeister moves are the following (I shall use the same names as Polyak [19], but with the letter Ω replaced by the letter R ): - -four oriented Reidemeister moves of type 1: - -four oriented Reidemeister moves of type 2: <details> <summary>Image 63 Details</summary>  ### Visual Description \n ## Diagram: Representation of Spiral Patterns ### Overview The image presents a 2x2 grid of diagrams depicting different spiral-like patterns, each labeled with a unique identifier: R1a, R1b, R1c, and R1d. Each diagram shows a line with a curved or looped section, and a bidirectional arrow positioned horizontally beneath it. The diagrams appear to illustrate variations in the shape and direction of a spiral or coiled structure. ### Components/Axes The diagram consists of the following components: * **Spiral/Coil Lines:** Black lines exhibiting different curved or looped shapes. * **Bidirectional Arrows:** Blue arrows positioned below each line, indicating a back-and-forth or oscillating movement. * **Labels:** Blue text labels (R1a, R1b, R1c, R1d) identifying each diagram. There are no explicit axes or scales present in the image. ### Detailed Analysis or Content Details The diagrams can be described as follows: * **R1a (Top-Left):** A line starts vertically, curves into a tight spiral, and then continues vertically upwards. The bidirectional arrow is positioned directly below the line. * **R1b (Top-Right):** A line starts vertically, curves slightly to the right, and then continues vertically upwards. The bidirectional arrow is positioned directly below the line. * **R1c (Bottom-Left):** A line starts vertically, curves into a loose spiral, and then continues vertically upwards. The bidirectional arrow is positioned directly below the line. * **R1d (Bottom-Right):** A line starts vertically, curves into a tight spiral, and then continues vertically upwards. The bidirectional arrow is positioned directly below the line. ### Key Observations The diagrams demonstrate variations in the tightness and shape of the spiral or coiled sections of the lines. The bidirectional arrows suggest an oscillating or reversible process associated with each pattern. The lines all start and end in a vertical direction. ### Interpretation The diagram likely represents different states or configurations of a dynamic system exhibiting spiral or coiled behavior. The labels (R1a, R1b, R1c, R1d) could represent different experimental conditions, time points, or stages in a process. The bidirectional arrows suggest that the system can transition between these states or configurations. Without further context, it is difficult to determine the specific meaning of these patterns. The image could be related to fluid dynamics, biological systems (e.g., DNA coiling), or other physical phenomena involving spiral structures. The consistent vertical start and end points suggest a defined input and output for the system being modeled. The variations in spiral tightness could represent different energy levels or degrees of stability within the system. </details> <details> <summary>Image 64 Details</summary>  ### Visual Description \n ## Diagram: DNA Strand Configuration Changes ### Overview The image depicts a diagram illustrating changes in the configuration of DNA strands. It shows four distinct states (R2a, R2b, R2c, R2d) representing transitions between a coiled/supercoiled DNA structure and a relaxed, linear structure. Each state is represented by a pair of DNA strand depictions with a bidirectional arrow indicating a reversible change. ### Components/Axes The diagram consists of four pairs of DNA strand configurations labeled R2a, R2b, R2c, and R2d. Each pair shows a transition between a coiled DNA structure (resembling a double helix) and a linear DNA structure. The arrows between the configurations are bidirectional, indicating reversibility. There are no axes or scales present. ### Detailed Analysis or Content Details The diagram presents four states: * **R2a:** Shows a transition between a coiled DNA strand (left) and a linear DNA strand (right). The coiled strand appears to have approximately 6-7 visible coils. * **R2b:** Shows a transition between a linear DNA strand (left) and a coiled DNA strand (right). The coiled strand appears to have approximately 6-7 visible coils. * **R2c:** Shows a transition between a coiled DNA strand (left) and a linear DNA strand (right). The coiled strand appears to have approximately 6-7 visible coils. * **R2d:** Shows a transition between a linear DNA strand (left) and a coiled DNA strand (right). The coiled strand appears to have approximately 6-7 visible coils. Each DNA strand depiction includes an arrow at one end, suggesting directionality or a specific end of the strand. The strands are depicted in black. ### Key Observations The diagram illustrates a reversible process of DNA strand coiling and relaxation. The four states (R2a-R2d) appear to represent different points in this process. The consistent number of coils in the coiled DNA structures suggests a relatively stable level of supercoiling. ### Interpretation The diagram likely represents the dynamic nature of DNA structure. DNA can exist in different conformations, including relaxed (linear) and supercoiled forms. These conformations are influenced by factors such as DNA topology, enzymatic activity (e.g., topoisomerases), and environmental conditions. The bidirectional arrows suggest that the transitions between these states are reversible, reflecting the ability of DNA to adapt to changing conditions. The labels R2a, R2b, R2c, and R2d could represent specific experimental conditions or stages in a biological process involving DNA conformational changes. Without further context, it is difficult to determine the precise meaning of these labels. The diagram is a simplified representation of a complex biological phenomenon. It does not provide quantitative data or specific details about the mechanisms driving these conformational changes. </details> -eight oriented Reidemeister moves of type 3: <details> <summary>Image 65 Details</summary>  ### Visual Description \n ## Diagram: Reaction Mechanism - R3 Series ### Overview The image depicts a series of chemical reaction steps, labeled R3a through R3h. Each step shows a transformation of a molecule represented by lines and diagonal lines with arrowheads, indicating bond changes. The diagram illustrates a reaction mechanism, likely involving a cyclic transition state. ### Components/Axes The diagram consists of eight pairs of molecular structures. Each pair is connected by a blue, bidirectional arrow labeled with "R3" followed by a letter (a-h). The structures within each pair represent the reactant and product of a single reaction step. There are no explicit axes or scales. The diagram is arranged in a 4x2 grid. ### Detailed Analysis or Content Details Each reaction step can be described as follows: * **R3a:** The initial structure has a horizontal line with a diagonal line crossing it from the top-left to the bottom-right, and another diagonal line crossing from the top-right to the bottom-left. Arrowheads are present on both diagonal lines pointing outwards. The product has the same horizontal line, but the diagonal lines have swapped positions and the arrowheads are now pointing inwards. * **R3b:** The initial structure has a horizontal line with a diagonal line crossing it from the top-left to the bottom-right, and another diagonal line crossing from the top-right to the bottom-left. Arrowheads are present on both diagonal lines pointing outwards. The product has the same horizontal line, but the diagonal lines have swapped positions and the arrowheads are now pointing inwards. * **R3c:** The initial structure has a horizontal line with a diagonal line crossing it from the top-left to the bottom-right, and another diagonal line crossing from the top-right to the bottom-left. Arrowheads are present on both diagonal lines pointing outwards. The product has the same horizontal line, but the diagonal lines have swapped positions and the arrowheads are now pointing inwards. * **R3d:** The initial structure has a horizontal line with a diagonal line crossing it from the top-left to the bottom-right, and another diagonal line crossing from the top-right to the bottom-left. Arrowheads are present on both diagonal lines pointing outwards. The product has the same horizontal line, but the diagonal lines have swapped positions and the arrowheads are now pointing inwards. * **R3e:** The initial structure has a horizontal line with a diagonal line crossing it from the top-left to the bottom-right, and another diagonal line crossing from the top-right to the bottom-left. Arrowheads are present on both diagonal lines pointing outwards. The product has the same horizontal line, but the diagonal lines have swapped positions and the arrowheads are now pointing inwards. * **R3f:** The initial structure has a horizontal line with a diagonal line crossing it from the top-left to the bottom-right, and another diagonal line crossing from the top-right to the bottom-left. Arrowheads are present on both diagonal lines pointing outwards. The product has the same horizontal line, but the diagonal lines have swapped positions and the arrowheads are now pointing inwards. * **R3g:** The initial structure has a horizontal line with a diagonal line crossing it from the top-left to the bottom-right, and another diagonal line crossing from the top-right to the bottom-left. Arrowheads are present on both diagonal lines pointing outwards. The product has the same horizontal line, but the diagonal lines have swapped positions and the arrowheads are now pointing inwards. * **R3h:** The initial structure has a horizontal line with a diagonal line crossing it from the top-left to the bottom-right, and another diagonal line crossing from the top-right to the bottom-left. Arrowheads are present on both diagonal lines pointing outwards. The product has the same horizontal line, but the diagonal lines have swapped positions and the arrowheads are now pointing inwards. All reactions appear to be identical in terms of structural change. ### Key Observations The diagram shows a series of eight identical reaction steps (R3a-R3h). Each step involves a rearrangement of the diagonal lines and a reversal of the direction of the arrowheads. This suggests a reversible reaction or a series of steps within a larger cyclic process. ### Interpretation The diagram likely represents a series of concerted, reversible steps in a chemical reaction mechanism. The consistent transformation across all R3 steps suggests a repeating motif or a series of equivalent transition states. The arrowheads indicate the direction of electron flow or bond formation/breaking. The overall process could be part of a larger catalytic cycle or a series of equilibrium reactions. Without further context, it's difficult to determine the specific chemical reaction being depicted, but the diagram clearly illustrates a mechanistic pathway involving a structural rearrangement. The repetition of the same transformation in each step suggests a highly ordered and potentially efficient process. </details> Crossings from emergent algebras. In section 5, example 7, it is mentioned that there is a connection between tangle diagrams and emergent algebras, via the notion of a quandle. Quandles are self-distributive idempotent quasigroups, which were invented as decorations of the arrows of a tangle diagram, which are invariant with respect to the Reidemeister moves. Let us define the emergent algebra crossing macros. (We can choose to neglect the ε decorations of the crossings, or, on the contrary, we can choose to do like in definition 5.5 of the emergent algebra sector, namely to add a set A to the group Γ and use even more nuanced decorations for the crossings.) <details> <summary>Image 66 Details</summary>  ### Visual Description \n ## Diagram: Feynman Diagrams - Particle Interactions ### Overview The image presents two pairs of Feynman diagrams illustrating particle interactions. Each pair shows a diagram in a more complex form on the left, and its simplified equivalent on the right. The diagrams depict interactions involving particles and their antiparticles, represented by lines with arrows indicating direction of propagation. The diagrams use specific symbols for vertices and internal lines. ### Components/Axes The diagrams consist of the following components: * **Lines with Arrows:** Represent particles and antiparticles. The direction of the arrow indicates the direction of propagation (forward in time). * **Vertices:** Points where particles interact. These are represented by circles with symbols inside. * **Internal Lines:** Lines connecting vertices, representing the exchange of virtual particles. * **Symbol 'ε':** Appears on internal lines, likely representing a specific type of virtual particle (e.g., a photon). * **Symbol 'Y':** Appears inside a vertex, likely representing a specific type of interaction. * **Equal Sign (===):** Indicates equivalence between the complex and simplified diagrams. ### Detailed Analysis or Content Details **Diagram 1 (Top Row):** * **Left Side:** Two incoming lines converge at a vertex containing the 'Y' symbol. One line goes down, labeled with 'ε', and the other continues as an outgoing line. Two outgoing lines emerge from the bottom vertex, labeled with 'ε'. * **Right Side:** Two lines cross, with an internal line labeled 'ε' connecting them. The lines are curved, indicating interaction. **Diagram 2 (Bottom Row):** * **Left Side:** Two incoming lines cross, with an internal line labeled 'ε' connecting them. The lines converge at a vertex containing the 'Y' symbol. One line goes down, labeled with 'ε', and the other continues as an outgoing line. Two outgoing lines emerge from the bottom vertex, labeled with 'ε'. * **Right Side:** Two lines cross, with an internal line labeled 'ε' connecting them. The lines are curved, indicating interaction. ### Key Observations * The diagrams demonstrate a simplification process in Feynman diagram representation. The complex diagrams on the left are equivalent to the simpler diagrams on the right. * The 'ε' symbol consistently appears on internal lines, suggesting the exchange of a specific virtual particle. * The 'Y' symbol appears at the vertices, indicating a specific type of interaction. * The diagrams are symmetrical in their structure, with the simplified versions representing the core interaction. ### Interpretation These diagrams likely represent a fundamental interaction in particle physics, such as an electromagnetic interaction. The 'ε' symbol likely represents a photon, the carrier of the electromagnetic force. The 'Y' symbol could represent a specific type of vertex, such as an electron-positron annihilation or creation. The simplification process demonstrates that complex interactions can be represented in a more concise and intuitive manner using Feynman diagrams. The diagrams illustrate the concept of virtual particles mediating interactions between real particles. The equivalence shown by the "===" sign suggests that the simplified diagram captures the essential physics of the more complex one, making calculations easier while preserving the underlying physical process. The diagrams are a visual representation of quantum electrodynamics (QED) or a similar quantum field theory. </details> In [6], sections 3-6 are dedicated to the use of these crossings for exploring emergent algebras and spaces with dilations. All constructions and reasonings from there can be put into the graphic lambda calculus formalism. Here I shall explain only some introductory facts. Let us associate to any locally planar tangle diagram T a graph in [ T ] ∈ GRAPH , called the translation of T , which is obtained by replacing the crossings with the emergent crossing macros (for a fixed ε ). Also, to any Reidemeister move we associate it's translation in graphic lambda calculus, consisting in a local move between the translations of the LHS and RHS tangles which appear in the respective move. (Note: these translations are not added to the moves which define graphic lambda calculus.) Theorem 6.1 The translations of all oriented Reidemeister moves of type 1 and 2 can be realized as sequences of the following moves from graphic lambda calculus: emergent algebra moves (R1a, R1b, R2, ext2), fan-out moves (i.e. CO-COMM, CO-ASSOC, global FANOUT) and pruning moves. More precisely the translations of the Reidemeister moves R1a, R1b are, respectively, the graphic lambda calculus moves R1a, R1b, modulo fan-out moves. Moreover, all translations of Reidemeister moves of type 2 can be expressed in graphic lambda calculus with the move R2, fan-out and pruning moves. The proof is left to the interested reader, see however section 3.4 [6]. The fact that the Reidemeister moves of type 3 are not true for (the algebraic version of) the emergent algebras, i.e. that the translations of those cannot be expressed as a sequence of moves from graphic lambda calculus, is a feature of the formalism and not a weakness. This is explained in detail in sections 5, 6 [6], but unfortunately at the moment of the writing that article the graphic lambda calculus was not available. It is an interesting goal the one of expressing the constructions from the mentioned sections as statements about the computability in the sense of definition 5.6 of the translations of certain tangle diagrams. As a justification for this point of view, let us remark that all tangle diagrams which appear in the Reidemeister moves of type 3 have translations which are related to the approximate difference or approximate sum graphs from definition 5.4. For example, let's take the translation of the graph from the RHS of the move R3d and call it D . This graph has three inputs and three outputs. Let's then consider a graph formed by grafting three graphs A , B , C at the inputs of D , such that A , B , C are not otherwise connected. Then we can perform the following sequence of moves. <details> <summary>Image 67 Details</summary>  ### Visual Description \n ## Diagram: Network Transformation ### Overview The image presents a series of diagrams illustrating a transformation of a network represented by lines and numbers into a more complex network of nodes and edges. The diagrams demonstrate a concept of "co-assoc" and "global fan-out" relating to network structure. The initial network is shown on the left, and its equivalent representation is shown on the right. Two further transformations are shown below, with arrows indicating the relationship between "global" and "fan-out". ### Components/Axes The diagrams consist of: * **Lines:** Representing connections or relationships between elements (A, B, C). * **Numbers (1, 2, 3):** Labels associated with the lines, likely indicating some form of weighting or identification. * **Nodes:** Represented by circles with the symbol "ε" inside. * **Arrows:** Indicating the direction of relationships or transformations. * **Labels:** "A", "B", "C" identifying network elements. * **Textual Annotations:** "CO-ASSOC", "GLOBAL", "FAN-OUT". ### Detailed Analysis or Content Details **Top Row:** * **Left Diagram:** Three lines are shown. Line B is horizontal and labeled with "3". Line C is angled downwards and labeled with "2". Line A is angled upwards and labeled with "1". The lines intersect. * **Right Diagram:** This diagram represents the equivalent network of the left diagram. It consists of nodes connected by edges. * Line B (labeled "3") is represented by a node with "ε" connected to a node with "A". * Line C (labeled "2") is represented by a node with "ε" connected to a node with "A". * Line A (labeled "1") is represented by a node with "ε" connected to a node with "A". * There are connections between the "ε" nodes. **Bottom Row:** * **Left Diagram:** A more complex network with nodes and edges. * Line C (labeled "2") enters from the top-left and connects to a node with "ε". * Line B (labeled "3") enters from the top-right and connects to a node with "ε". * Line A (labeled "1") exits from the bottom-right and connects to a node with "ε". * There are multiple connections between the "ε" nodes. * **Right Diagram:** Another network representation. * Line C (labeled "2") enters from the top-left and connects to a node with "ε". * Line B (labeled "3") enters from the top-right and connects to a node with "ε". * Line A (labeled "1") exits from the bottom-right and connects to a node with "ε". * There are multiple connections between the "ε" nodes. **Annotations:** * A bidirectional arrow labeled "CO-ASSOC" is positioned between the top two diagrams. * A bidirectional arrow labeled "GLOBAL <-> FAN-OUT" is positioned between the bottom two diagrams. ### Key Observations * The diagrams demonstrate a transformation from a simple line-based network to a more complex node-and-edge network. * The numbers (1, 2, 3) appear to be preserved during the transformation, acting as identifiers. * The "ε" symbol consistently appears within the nodes, suggesting it represents a common element or operation. * The "CO-ASSOC" annotation suggests a relationship of co-association between the initial line network and its node-based equivalent. * The "GLOBAL <-> FAN-OUT" annotation indicates a relationship between the two bottom diagrams, potentially representing a global view versus a fan-out distribution. ### Interpretation The diagrams likely illustrate a method for representing network relationships in a more detailed and structured manner. The initial line network is a simplified representation, while the node-and-edge network provides a more granular view of the connections and interactions. The "ε" symbol could represent a fundamental unit of interaction or a transformation operation. The "CO-ASSOC" annotation suggests that the node-and-edge network is a faithful representation of the relationships present in the original line network. The "GLOBAL <-> FAN-OUT" annotation suggests that the network can be viewed from a global perspective or broken down into a fan-out distribution, potentially for analysis or optimization. The diagrams are abstract and do not provide specific data values, but rather demonstrate a conceptual transformation of network representation. The diagrams are likely part of a theoretical framework for network analysis or design. </details> The graph from the left lower side is formed by an approximate difference, a ¯ ε gate and several Υ gates. Therefore, if A , B , C are computable in the sense of definition 5.4 then the initial graph (the translation of the LHS of R3d with A , B , C grafted at the inputs) is computable too. Graphic beta move as braiding. Let us now construct crossings, in the sense previously explained, from gates coming from lambda calculus. <details> <summary>Image 68 Details</summary>  ### Visual Description \n ## Diagram: Signal Processing Network ### Overview The image presents two diagrams illustrating signal processing networks, likely related to modulation or mixing of signals. Each diagram consists of a network of lines representing signal flow, with circular nodes indicating operations on the signals. The diagrams are paired with equivalent representations using standard signal flow diagrams. ### Components/Axes The diagrams contain the following components: * **Signal Input/Output Lines:** Arrows indicate the direction of signal flow. * **Circular Nodes:** * A circle with a symbol "λ" inside, likely representing a multiplier or modulator. * A circle with a "Y" shape inside, likely representing a summing junction. * **Equal Sign:** "===" indicates equivalence between the network diagram and the standard signal flow diagram. ### Detailed Analysis or Content Details **Diagram 1 (Top):** * Two input signals enter the network. * One signal passes through the modulator (λ). * Both signals are fed into the summing junction (Y). * The output of the summing junction is represented by a waveform. * The equivalent signal flow diagram shows two input signals, one passing through a multiplier, and then both signals being summed. **Diagram 2 (Bottom):** * Two input signals enter the network. * One signal passes through the summing junction (Y). * The output of the summing junction is fed into the modulator (λ). * The output of the modulator is represented by a waveform. * The equivalent signal flow diagram shows two input signals, one being summed, and then the sum being multiplied. ### Key Observations The diagrams demonstrate the impact of the order of operations (modulation vs. summation) on the resulting signal. The diagrams are visually symmetrical, with the only difference being the placement of the modulator and summing junction. ### Interpretation These diagrams illustrate fundamental concepts in signal processing, specifically how the order of operations affects the outcome. The first diagram shows a signal being modulated *after* being combined with another signal. The second diagram shows a signal being modulated *before* being combined. This difference can have significant implications in applications like amplitude modulation (AM) or frequency modulation (FM) where the order of mixing and summing determines the characteristics of the modulated signal. The diagrams are not providing numerical data, but rather a conceptual illustration of signal flow and processing. The use of the "===" symbol suggests a mathematical equivalence between the network representation and the standard signal flow diagram, implying that the two representations are interchangeable for analysis purposes. </details> As previously, we define translations of (locally planar) tangle diagrams into graphs in GRAPH . The class of locally planar tangle diagrams is out in a one-to one correspondence with a class of graphs in GRAPH , let us call this class λ -TANGLE . We could proceed in the inverse direction, namely consider the class of graphs λ -TANGLE , along with the moves: graphic beta move and elimination of loops. Then we make the (inverse) translation of graphs in λ -TANGLE into locally planar tangle diagrams and the (inverse) translation of the graphic beta move and the elimination of loops. The following proposition explains what we obtain. Proposition 6.2 The class of graphs λ -TANGLE is closed with respect to the application of the graphic beta move and of the elimination of loops. The translations of the graphic beta and elimination of loops moves are the following SPLICE 1, 2 (translation of the graphic beta move) and LOOP 1, 2 (translation of the elimination of loops) moves. Proof. The proposition becomes obvious if we find the translation of the graphic beta move. There is one translation for each crossing. (Likewise, there are two translations for elimination of loops, depending on the orientation of the loop which is added/erased.) <details> <summary>Image 69 Details</summary>  ### Visual Description \n ## Diagram: Splice and Loop Operations on Tangled Lines ### Overview The image depicts a series of diagrams illustrating operations on tangled lines, specifically "Splice" and "Loop" operations. Each operation takes a configuration of intersecting lines (a "Tangle Diagram") as input and transforms it into a different configuration of lines. The diagrams are arranged in two rows, with "Splice" operations in the top row and "Loop" operations in the bottom row. ### Components/Axes The diagram consists of the following components: * **Tangle Diagrams:** Irregularly shaped blobs labeled "TANGLE DIAGRAM" representing configurations of intersecting lines. * **Lines:** Straight lines with arrowheads indicating direction. * **Labels:** Text labels indicating the operation being performed ("SPLICE 1", "SPLICE 2", "LOOP 1", "LOOP 2"). * **Arrows:** Bidirectional arrows connecting the input and output tangle diagrams, indicating the transformation. * **Circles:** Small circles on the right side of the bottom two diagrams. ### Detailed Analysis or Content Details The diagram is structured into four distinct transformations: 1. **SPLICE 1:** * Input: A tangle diagram with two lines crossing each other. Both lines have arrowheads pointing towards the intersection. * Transformation: Indicated by a bidirectional arrow labeled "SPLICE 1". * Output: A tangle diagram with two parallel lines, both with arrowheads pointing in the same direction. 2. **SPLICE 2:** * Input: A tangle diagram with two lines crossing each other. Both lines have arrowheads pointing towards the intersection. * Transformation: Indicated by a bidirectional arrow labeled "SPLICE 2". * Output: A tangle diagram with two parallel lines, both with arrowheads pointing in the same direction. 3. **LOOP 1:** * Input: A tangle diagram with two lines crossing each other. Both lines have arrowheads pointing towards the intersection. * Transformation: Indicated by a bidirectional arrow labeled "LOOP 1". * Output: A tangle diagram with two lines, and a small circle on the right. 4. **LOOP 2:** * Input: A tangle diagram with two lines crossing each other. Both lines have arrowheads pointing towards the intersection. * Transformation: Indicated by a bidirectional arrow labeled "LOOP 2". * Output: A tangle diagram with two lines, and a small circle on the right. ### Key Observations * The "Splice" operations appear to untangle the lines, resulting in parallel lines. * The "Loop" operations also start with a tangle diagram, but the output includes a small circle, suggesting the creation of a closed loop. * The input tangle diagrams are identical for all four operations. * The outputs of "SPLICE 1" and "SPLICE 2" are identical. * The outputs of "LOOP 1" and "LOOP 2" are identical. ### Interpretation The diagram illustrates two distinct operations – "Splice" and "Loop" – that can be applied to tangled line configurations. The "Splice" operation seems to resolve the tangle by aligning the lines, while the "Loop" operation introduces a circular element. The fact that the input is the same for all operations, and that SPLICE 1 and 2, and LOOP 1 and 2 produce the same outputs, suggests that these operations might be deterministic, or that the diagram is only showing one possible outcome for each operation. The small circles in the "Loop" diagrams likely represent a closed loop or a change in the topological structure of the lines. This could be a visual representation of mathematical concepts related to knot theory or braid groups, where "splices" and "loops" are fundamental operations. The diagram doesn't provide quantitative data, but rather a qualitative illustration of transformations. </details> <details> <summary>Image 70 Details</summary>  ### Visual Description \n ## Diagram: Splice Diagrams ### Overview The image presents two sets of diagrams illustrating a concept related to splices, likely in the context of signal processing or genetics. Each set consists of two diagrams separated by three vertical lines ("==="), representing an equivalence or transformation. The diagrams depict circular nodes labeled "λ" and "β", connected by arrows representing flow or relationships. Horizontal lines with arrowheads represent linear sequences. The right-hand diagrams in each set are labeled "SPLICE 1" and "SPLICE 2" respectively. ### Components/Axes The diagrams contain the following components: * **Circular Node (λ):** A circle with the label "λ" inside. * **Circular Node (β):** A circle with the label "β" inside. * **Arrows:** Curved and straight arrows indicating direction of flow or connection. * **Horizontal Lines with Arrowheads:** Representing linear sequences. * **Labels:** "λ", "β", "SPLICE 1", "SPLICE 2". * **Equivalence Indicators:** Three vertical lines ("===") separating diagrams. ### Detailed Analysis or Content Details **Set 1:** * **Left Diagram:** A circular node labeled "λ" is at the center. Two arrows enter the node from the top-left and top-right, and two arrows exit from the bottom-left and bottom-right. An arrow labeled "β" curves from the bottom-right exit arrow back to the top-left input arrow. Two horizontal lines with arrowheads are below the circular node, with arrows pointing to the right. * **Right Diagram:** Two arrows enter from the top-left and top-right, and two arrows exit from the bottom-left and bottom-right. An arrow labeled "SPLICE 1" curves from the bottom-right exit arrow back to the top-left input arrow. Two horizontal lines with arrowheads are below, with arrows pointing to the right. **Set 2:** * **Left Diagram:** A circular node labeled "λ" is at the center. Two arrows enter the node from the top-left and top-right, and two arrows exit from the bottom-left and bottom-right. An arrow labeled "β" curves from the bottom-right exit arrow back to the top-left input arrow. Two horizontal lines with arrowheads are below the circular node, with arrows pointing to the right. * **Right Diagram:** Two arrows enter from the top-left and top-right, and two arrows exit from the bottom-left and bottom-right. An arrow labeled "SPLICE 2" curves from the bottom-right exit arrow back to the top-left input arrow. Two horizontal lines with arrowheads are below, with arrows pointing to the right. ### Key Observations The diagrams in each set are visually similar, with the primary difference being the label on the curved arrow: "β" in the left diagrams and "SPLICE 1" or "SPLICE 2" in the right diagrams. The "===" symbols suggest that the left and right diagrams in each set are equivalent representations of the same process, with the curved arrow representing a splicing operation. ### Interpretation The diagrams likely represent a splicing process where a circular element (λ) modifies or connects linear sequences. The "β" label might represent a general splicing factor or process, while "SPLICE 1" and "SPLICE 2" represent specific splicing events or variations. The equivalence indicated by the "===" suggests that the splicing operation (β) can be represented equivalently by the modified diagram with the "SPLICE" label. The diagrams are abstract and do not provide quantitative data, but they illustrate a conceptual relationship between a circular element, linear sequences, and a splicing process. The diagrams could be used to explain different splicing scenarios or to illustrate the effect of a splicing factor on a particular sequence. The diagrams are not providing facts or data, but rather a visual representation of a process. </details> The following theorem clarifies which are the oriented Reidemeister moves which can be expressed as sequences of graphic lambda calculus moves applied to graphs in λ -TANGLE . Among these moves, some are more powerful than others, as witnessed by the following Theorem 6.3 All the translations of the oriented Reidemeister move into moves between graphs in λ -TANGLE , excepting R2c, R2d, R3a, R3h, can be realized as sequences of graphic beta moves and elimination of loops. Moreover, the translations of moves R2c, R2d, R3a, R3h are equivalent up to graphic beta moves and elimination of loops (i.e. any of these moves, together with the graphic beta move and elimination of loops, generates the other moves from this list). Proof. It is easy, but tedious, to verify that all the mentioned moves can be realized as sequences of SPLICE and LOOP moves. It is as well easy to verify that the moves R2c, R2d, R3a, R3h are equivalent up to SPLICE and LOOP moves. It is not obvious that the moves R2c, R2d, R3a, R3h can't be realized as a sequence of SPLICE and LOOP moves. In order to do this, we prove that R2d can't be generated by SPLICE and LOOP. Thanks are due to Peter Kravchuk for the idea of the proof, given in an answer to a question I asked on mathoverflow [7], where I described the moves SPLICE and LOOP. To any locally planar tangle diagram A associate it's reduced diagram R(A), which is obtained by the following procedure: first use SPLICE 1,2 from left to right for all crossings, then use LOOP 1,2 from right to left in order to eliminate all loops which are present at this stage. Notice that: -the order of application of the SPLICE moves does not matter, because they are applied only once per crossing. There is a finite number of splices, equal to the number of crossings. Define the bag of splices SPLICE(A) to be the set of SPLICE moves applied. - -The same is true for the order of eliminations of loops by LOOP 1, 2. There is a finite number of loop eliminations, because the number of loops (at this stage) cannot be bigger than the number of edges of the initial diagram. Define the bag of loops LOOP(A) to be the set of all loops which are present after all splices are done. Let us now check that the reduced diagram does not change if one of the 4 moves is applied to the initial diagram. Apply a SPLICE 1,2 move to the initial diagram A, from left to right, and get B. Then SPLICE(B) is what is left in the bag SPLICE(A) after taking out the respective splice. Also LOOP(B) = LOOP(A) because of the definition of bags of loops. Therefore R(A) = R(B). Apply a SPLICE 1, 2 from right to left to A and get B. Then R(A) = R(B) by the same proof, with A, B switching places. Apply a LOOP1, 2 from left to right to A and get B. The new loop introduced in the diagram does not participate to any crossing (therefore SPLICE(A) = SPLICE(B)), so we find it in the bag of loops of B, which is made by all the elements of LOOP(A) and this new loop. Therefore R(A) = R(B). Same goes for LOOP1, 2 applied from right to left. Finally, remark that the reduced diagram of the LHS of the move R2d is different than the reduced diagram of the RHS of the move R2d, therefore the move R2d cannot be achieved with a sequence of splices and loops addition/elimination. ## References - [1] A. Bella¨ ıche, The tangent space in sub-Riemannian geometry, in: Sub-Riemannian Geometry, A. Bella¨ ıche, J.-J. Risler eds., Progress in Mathematics , 144 , Birkh¨ auser, (1996), 4-78 - [2] M. Buliga, Dilatation structures I. Fundamentals, J. Gen. Lie Theory Appl. , 1 (2007), 2, 65-95. arXiv:math/0608536 - [3] M. Buliga, Emergent algebras, arXiv:0907.1520 - [4] M. Buliga, Braided spaces with dilations and sub-riemannian symmetric spaces, in: Geometry. Exploratory Workshop on Differential Geometry and its Applications, eds. D. Andrica, S. Moroianu, Cluj-Napoca 2011, 21-35, arXiv:1005.5031 - [5] M. Buliga, λ -Scale, a lambda calculus for spaces with dilations, arXiv:1205.0139 - [6] M. 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